Angular Momentum Equation for Fluid Systems

Understanding how forces cause rotation is fundamental in mechanics, but when the system is a flowing fluid—like water in a pump or steam in a turbine—the analysis becomes more complex. The angular momentum equation for fluid systems, often called the moment of momentum equation, is the powerful tool that bridges this gap. It allows engineers to analyze and design the rotating machinery that powers our world, from the hydroelectric turbines generating electricity to the compressors in jet engines, by directly relating the torque applied to the change in the swirling motion of a fluid.

From Newton’s Law to Fluid Control Volumes

The principle originates from Newton’s second law for rotation, which states that the sum of external torques ($\sum \vec{T}$) acting on a system equals the time rate of change of its angular momentum ($\vec{H}$). For a system of particles, angular momentum about a point O is defined as ${\vec{H}}_O={\int }_{sys}\vec{r}\times \vec{V}dm$, where $\vec{r}$ is the position vector and $\vec{V}$ is the velocity.

To apply this law to a fluid, which continuously deforms, we use the Reynolds Transport Theorem (RTT). The RTT lets us track how a property (here, angular momentum) changes for a control volume (CV)—a fixed region in space we choose for analysis. The theorem states: $$\frac{d{\vec{H}}_{sys}}{dt}=\frac{\partial }{\partial t}{\int }_{CV}(\vec{r}\times \vec{V})\rho dV+{\int }_{CS}(\vec{r}\times \vec{V})\rho \vec{V}\cdot d\vec{A}$$ This translates Newton’s law into a practical, analyzable form: the left side, $d{\vec{H}}_{sys}/dt$, equals the net external torque. Therefore, the moment of momentum equation for a control volume is: $$\sum {\vec{T}}_{CV}=\frac{\partial }{\partial t}{\int }_{CV}(\vec{r}\times \vec{V})\rho dV+{\int }_{CS}(\vec{r}\times \vec{V})\rho \vec{V}\cdot d\vec{A}$$ The term $\sum {\vec{T}}_{CV}$ includes all external torques: shaft torque, and torques due to pressure and shear forces on the control surface. The first integral on the right represents the rate of change of angular momentum within the CV, while the second (surface) integral represents the net flux of angular momentum out of the CV.

The Steady-Flow, One-Dimensional Form for Turbomachinery

For the vast majority of turbomachinery applications—pumps, turbines, fans, and compressors—analysis uses a simplified, powerful version of the equation. We assume steady flow (no change within the CV over time) and one-dimensional flow at the inlet and outlet, meaning properties are uniform across the entry and exit cross-sections.

Under these conditions, the unsteady integral term vanishes, and the flux integral simplifies. The equation reduces to a scalar form about the axis of rotation (often the z-axis): $$T_{shaft}=\dot{m}(r_2{V}_{\theta ,2}-r_1{V}_{\theta ,1})$$ Where:

• $T_{shaft}$ is the net shaft torque exerted on the contents of the CV.
• $\dot{m}$ is the mass flow rate.
• $r_1$ and $r_2$ are the radial distances from the axis to the inlet and outlet, respectively.
• $V_{\theta ,1}$ and $V_{\theta ,2}$ are the swirl, or tangential, components of the absolute fluid velocity at the inlet and outlet.

This is the workhorse equation for preliminary turbomachinery design. The sign convention is crucial: torque and power are positive when energy is added to the fluid (as in a pump), and negative when extracted from the fluid (as in a turbine).

Power, Head, and the Euler Turbomachine Equation

The power transferred between the shaft and the fluid is simply the torque multiplied by the rotational speed ($\omega$): ${\dot{W}}_{shaft}=T_{shaft}\omega =\dot{m}\omega (r_2{V}_{\theta ,2}-r_1{V}_{\theta ,1})$.

Since $\omega r$ is the impeller or blade speed $U$, we arrive at the famous Euler turbomachine equation: $${\dot{W}}_{shaft}=\dot{m}(U_2{V}_{\theta ,2}-U_1{V}_{\theta ,1})$$ For pumps and compressors, this represents the ideal head added to the fluid. The ideal head ($H_{ideal}$) is the mechanical energy per unit weight, found by dividing shaft power by $\dot{m}g$: $$H_{ideal}=\frac{1}{g}(U_2{V}_{\theta ,2}-U_1{V}_{\theta ,1})$$ This equation reveals the three primary means of energy transfer in rotating machinery: changes in blade speed ($U$), changes in the swirl component of absolute velocity ($V_{\theta }$), or both. In a centrifugal pump, for example, fluid enters with little swirl ($V_{\theta ,1}\approx 0$) at a small radius. The impeller blades increase both the radius ($r_1\to r_2$) and impart a tangential velocity, resulting in a large positive $U_2{V}_{\theta ,2}$ term and a significant increase in fluid head.

Applying the Equation: Analysis of a Simple Turbine

Let’s walk through a simplified analysis of an axial-flow turbine stage (like in a jet engine) to see the equation in action. Assume:

• Steam or gas enters a row of stationary guide vanes (stators), which impart swirl.
• The fluid then enters the rotating blades (rotors). Our control volume encloses just the rotor.
• Inlet to rotor: $V_{\theta ,1}$ is large and positive (in the direction of rotation).
• Design goal: Extract maximum work, meaning we want $U_2{V}_{\theta ,2}$ to be much smaller than $U_1{V}_{\theta ,1}$. Ideally, $V_{\theta ,2}\approx 0$ or even becomes slightly negative.
• Assume $U_1=U_2=U$ (axial machine), $\dot{m}$ is known, $V_{\theta ,1}$ is given.

Step 1: Calculate Torque. Using $T_{shaft}=\dot{m}(r{V}_{\theta ,2}-r{V}_{\theta ,1})=\dot{m}r(V_{\theta ,2}-V_{\theta ,1})$. Since $V_{\theta ,2}<V_{\theta ,1}$, the torque $T_{shaft}$ will be negative, indicating torque is exerted by the fluid on the rotor (power extraction).

Step 2: Calculate Power. ${\dot{W}}_{shaft}=T_{shaft}\omega =\dot{m}U(V_{\theta ,2}-V_{\theta ,1})$. Again, this is negative, representing useful power output.

Step 3: Relate to Pressure Drop. The extracted shaft power comes from a drop in the fluid's total pressure (or enthalpy) as it passes through the rotor. This direct link between kinematics ($V_{\theta }$) and thermodynamics is what makes the angular momentum equation so indispensable.

Common Pitfalls

1. Incorrect Sign for Torque and Power: The most frequent error is misinterpreting what is positive and negative work. Remember the equation $T_{shaft}=\dot{m}(r_2{V}_{\theta ,2}-r_1{V}_{\theta ,1})$ gives torque on the CV contents. For a pump, you put torque in, so $T_{shaft}$ (on the fluid) is positive, requiring $r_2{V}_{\theta ,2}>r_1{V}_{\theta ,1}$. For a turbine, the fluid exerts torque on the rotor, so $T_{shaft}$ is negative, meaning $r_2{V}_{\theta ,2}<r_1{V}_{\theta ,1}$.

1. Confusing Velocity Components: Failing to correctly identify the tangential velocity component $V_{\theta }$ is critical. You must use the absolute fluid velocity (relative to a stationary observer), not the velocity relative to the moving blade. These are related by the vector triangle: ${\vec{V}}_{absolute}={\vec{V}}_{blade}+{\vec{V}}_{relative}$.

1. Misapplying the Steady-Flow Form: Using the simplified $T_{shaft}=\dot{m}(r_2{V}_{\theta ,2}-r_1{V}_{\theta ,1})$ when the flow is unsteady or when properties are not uniform across the inlet/outlet will give incorrect results. It is valid only for steady flow with uniform one-dimensional inlets and exits.

1. Neglecting All External Torques: The equation $\sum {\vec{T}}_{CV}$ includes more than just shaft torque. In some analyses, especially for non-rotating CVs, torques from surface pressure and shear forces (e.g., on a pipe bend) must be accounted for. Overlooking these can lead to an unbalanced analysis.

Summary

• The angular momentum equation for a control volume is derived from Newton’s second law for rotation via the Reynolds Transport Theorem, equating net external torque to the rate of change of angular momentum within the CV plus the net flux out of it.
• For steady-flow turbomachinery analysis, it simplifies to a scalar equation: $T_{shaft}=\dot{m}(r_2{V}_{\theta ,2}-r_1{V}_{\theta ,1})$, which directly relates shaft torque to the change in the swirl of the fluid.
• This leads to the Euler turbomachine equation, ${\dot{W}}_{shaft}=\dot{m}(U_2{V}_{\theta ,2}-U_1{V}_{\theta ,1})$, which is the cornerstone for calculating ideal power transfer and head in pumps, turbines, fans, and compressors.
• The sign of torque and power is paramount: positive for energy addition to the fluid (pumps), negative for energy extraction (turbines).
• Successful application requires careful identification of a control volume, correct use of the absolute tangential velocity component ($V_{\theta }$), and adherence to the assumptions (steady, one-dimensional flow) when using the simplified form.