20 Conservation of Energy: Energy Equation & Bernoulli’s Equation

Introduction

Energy is defined as the ability to do work, which is a very useful property in engineering. There are many types of energy, including potential, kinetic, chemical, electrical, nuclear, etc. For thermodynamic analyses, energy can be classified into two categories: macroscopic energy and microscopic energy. Macroscopic energy is energy that a whole system possesses with respect to a fixed external reference, and in thermodynamics, these are potential and kinetic energies. Microscopic energy is the energy contained in the system at the molecular level. A system can be defined as a collection of matter of fixed identity.

When developing the energy equation for a fluid flow, the applicable physical principle is a thermodynamic one in that energy cannot be created or destroyed but only converted from one form to another. This principle is formally embodied in the first law of thermodynamics applied to a system of a given (fixed) mass, i.e.,

(1)   \begin{equation*} \Delta Q + \Delta W = \Delta E \end{equation*}

where \Delta Q is the change in the net heat added to the system (such as from a thermal source), \Delta W is the change in the net work done on the system (such as mechanical work), and \Delta E is the change in internal energy of the mass inside that system, the principle being illustrated in the schematic below. Notice that \Delta Q is positive when heat is added to the system and negative when heat is lost. Likewise, \Delta W is positive when work is done on the system and negative when the system does work.

 

The first law of thermodynamics states that the change in internal energy of a fixed system is equal to net heat added less the net work done on that system.

Internal energy can be considered the sum of the macroscopic kinetic and potential energy of the molecules comprising the fluid and their microscopic internal energy, e.g., temperature effects. Indeed, when dealing with thermodynamic relations, then for a perfect gas the internal energy is a function of temperature alone, i.e.,

(2)   \begin{equation*} D = f(T) \end{equation*}

Therefore, another way of writing the first law of thermodynamics for a fixed system is

(3)   \begin{equation*} ( Q_{\rm in} - Q_{\rm out} ) + ( W_{\rm in} - W_{\rm out} ) = \Delta E \end{equation*}

If the system does no external work, all the heat added will manifest as an increase in internal energy, e.g., an increase in temperature. Of course, heat and work can be added to or subtracted from an engineering system, i.e., they can be transferred across the system’s boundaries.

The system’s energy can be viewed as being associated with that system’s specific amount of mass. Therefore, for a fluid flow, the energy terms in the resulting equations are usually written as an energy per unit mass, i.e., per fluid element of  “\varrho \, d{\cal{V}}” in the spirit of the differential calculus. This approach is also consistent with the previous derivations of the equations that applied to the conservation of mass and momentum. If q is the heat added per unit mass and w is the work done per unit mass, then the conservation of energy in this differential form is

(4)   \begin{equation*} dq + dw = de \end{equation*}

Learning Objectives

  • Set up the most general form of the energy equation in integral form.
  • Know how to simplify the energy equation into various, more practical forms.
  • Understand how to derive a surrogate for the energy equation for steady, incompressible, inviscid flow, called the Bernoulli equation.
  • Learn how to solve some fundamental engineering problems using the energy equation.

Setting Up the Energy Equation

As before, consider a fluid flow through a fixed finite control volume {\cal V} bounded by a surface of area S, as shown in the figure below. The flow velocity is \vec{V} at any point inside the control volume or on the control surface. At a point on the control surface, the unit normal, \vec{n}, and the unit normal vector area, d\vec{S}, can be defined. Also, let d{\cal V} be an elemental fluid volume inside the control volume, which will contain a fluid of elemental mass \varrho \, d{\cal V}.

 

A finite control volume, fixed in space, is used to set up the energy equation in its most general form, i.e., as a governing equation.

The objective is to develop an analogous equation to the continuity and momentum equations for energy conservation as it applies to a fluid flow. As will be shown, however, the resulting energy equation describes a power balance equation for the fluid system, i.e., a balance of the rate of work done on the flow versus the rate of energy added to the flow and/or conversion from one form of energy in the flow to another.

Heat Addition

An increment in energy, de, can be added to the elemental fluid volume d{\cal{V}} in the form of heat, dq so that de = dq. Heat energy can be added (or subtracted) by conduction, convection, or radiation, e.g., because of a temperature difference. However, heat energy is often added to a system through combustion, e.g., the liberation of heat energy from stored chemical energy in a fuel. Therefore, the rate of volumetric heat addition will be

(5)   \begin{equation*} \mbox{Rate of volumetric heat addition} = \iiint_{\cal{V}} \overbigdot{q} \, \varrho \, d{\cal{V}} \end{equation*}

If the flow is viscous, then frictional effects can also add heat. This latter effect can be important on a high-speed (supersonic) aircraft or a spacecraft during re-entry into the Earth’s atmosphere. While viscous heating effects must be recognized as a complicated process involving the consideration of shear stresses and thermal effects from shock wave formation in a high-speed fluid flow, for now, the effects can be cumulatively denoted by \overbigdot{Q}_{\mu}. Therefore, for the volume of fluid, then

(6)   \begin{equation*} \mbox{Total rate of heat addition} = \iiint_{\cal{V}} \overbigdot{q} \, \varrho \, d {\cal{V}} + \overbigdot{Q}_{\mu} \end{equation*}

Caution: Symbol conflict is not uncommon between branches of engineering. In this case, Q is used for heat, and \overbigdot{Q} means the rate of heat addition. In other contexts, Q can mean volume flow rate. It is essential to avoid such conflicts, which can be done by redefining a new symbol. For example, the volume flow rate can be defined as \overbigdot{{\cal{V}}}.

Work

If some work is done on the system, i.e., dw per unit mass, the system’s energy will increase further. Work can be done by the effects of forces resulting from the action of pressure and/or body forces and/or viscous shear and/or mechanical work. Remember that work is equivalent to a force times a distance, and power is the rate of doing work; therefore, power is equivalent to the product of a force and a velocity.

In the case of pressure effects, a pressure force arises from the pressure acting over an area, i.e., for a point on the control surface then -p \, d\vec{S}, the minus sign indicating that the pressure force acts inward for an outward-pointing d\vec{S} (by convention). Therefore, following the same type of process used for the derivation of the momentum equation, the work done by the pressure forces can be expressed as

(7)   \begin{equation*} \mbox{Rate of doing work by pressure forces} = -\oiint_{S} (p \, d\vec{S}) \bigcdot \vec{V} \end{equation*}

Similarly, the effects of body forces can be written as

(8)   \begin{equation*} \mbox{Rate of doing work by body forces} = \iiint_{\cal{V}} ( \varrho \, \vec{f}_b \, d{\cal{V}} ) \bigcdot \vec{V} \end{equation*}

In the case of viscous work, this contribution can be written as \overbigdot{W}_{\mu} for the same reason as heat addition from viscous effects. This term is the rate of work done by viscous stresses acting on the control volume over the control surface.

Finally, some mechanical work could be added to the system, say \overbigdot{W}_{\rm mech}, an example being the work done by a pump. Work can also be extracted from a fluid flow, such as with a turbine, fan, or propeller. Therefore, adding all the contributions together gives

(9)   \begin{equation*} \mbox{Total rate of work} = -\oiint_{S} (p \, d\vec{S}) \bigcdot \vec{V} + \iiint_{\cal{V}} ( \varrho \, \vec{f}_b \, d{\cal{V}} ) \bigcdot \vec{V} + \overbigdot{W}_{\mu} + \overbigdot{W}_{\rm mech} \end{equation*}

Internal Energy

The next step is to examine the internal energy inside the control volume. Energy has the ability to do work, so this is an important part of the energy equation. This form of energy will comprise some “temperature” energy or the microscopic kinetic energy per unit mass of the fluid molecules, e, some macroscopic kinetic energy per unit mass of the fluid, V^2/2, and some macroscopic potential energy per unit mass, g z. In general, internal energy can be written as

(10)   \begin{equation*} \mbox{Energy per unit mass inside $d\cal{V}$} = e + \frac{V^2}{2} + g z \end{equation*}

where the resultant velocity V is obtained from V^2 = u^2 + v^2 + w^2.

Notice that the microscopic energy of the fluid molecules, e, can be considered as a sum of their translational kinetic, rotational, and vibrational energy, as suggested in the schematic below. The internal potential energy at the molecular level also contributes to total internal energy, e.g., molecular bonding, chemical, and nuclear energy. However, it is unnecessary to consider such latter effects at the present level of analysis. The best way of thinking about the internal energy of a fluid is in terms of its temperature.

The microscopic energy of the fluid molecules, e per unit mass, can be considered as the sum of their translational energy, rotational energy, and vibrational energy,

The temperature of the fluid, T, is proportional to the average total energy of the molecules, E. This molecular relationship is usually written as E = (3/2) k_B \, T, where k_B is known as Boltzmann’s constant. In this latter form, however, it strictly holds for monoatomic gases because they have three degrees of translational freedom, i.e., kinetic energy only. A thermal gradient, for example, will cause the molecules to move faster in the direction of increasing temperature. Thermal energy will be transferred from the hotter part of the gas to the cooler part, basically a statement of the second law of thermodynamics, i.e., energy flows “downhill.”

Diatomic gases, such as nitrogen and oxygen, which comprise about 98% of air, also have two degrees of freedom of rotational motion and two degrees of vibrational motion, the latter being important only at higher temperatures. Therefore, their total internal energy will be related using E = (5/2) k_B \, T at lower temperatures and E = (7/2) k_B \, T at higher temperatures.

The macroscopic kinetic energy per unit mass of the fluid per unit mass, V^2/2, and potential energy per unit mass, g z, can be interpreted within the framework of a continuum model, as shown in the schematic below. Increasing the velocity of the fluid volume or parcel will increase its kinetic energy, and increasing the height of the fluid or its pressure “head” will increase its potential energy. Some work on the fluid must be done to increase either, which requires energy.

The macroscopic kinetic and potential energy of the fluid can be interpreted within the framework of a continuum model.

Energy in & Out of Control Volume

The total energy will then be obtained by integration over the entire mass of fluid contained within the control volume, remembering that mass can also flow across the control surface from the control volume. Across the element dS, the mass flow rate will be \varrho \, \vec{V} \bigcdot d\vec{S}. Therefore, the total energy flow rate across the entire surface of the control volume then follows by integration, i.e.,

(11)   \begin{equation*} \mbox{Net flow of energy across $S$} = \oiint_{S} (\varrho \, \vec{V} \bigcdot d\vec{S}) \left( e + \frac{V^2}{2} + g z\right) \end{equation*}

In addition, unsteady effects may be present so that the energy in the system can change because of temporal variations of the flow field properties inside \cal{V}. For this contribution then, the total energy inside the control volume is

(12)   \begin{equation*} \mbox{Total energy inside $\cal{V}$} = \iiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z\right) d{\cal{V}} \end{equation*}

and so the time rate of change of energy inside the control volume will be

(13)   \begin{equation*} \mbox{Time rate of change of energy inside $\cal{V}$} = \frac{\partial}{\partial t} \iiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z\right) d{\cal{V}} \end{equation*}

Finally, adding these two latter terms together (Eqs. 11 and 13) gives

(14)   \begin{eqnarray*} \mbox{Total rate of change in energy} & = & \frac{\partial}{\partial t} \iiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z \right) d{\cal{V}} \\ && \nonumber \quad + \oiint_{S} (\varrho \, \vec{V} \bigcdot d\vec{S}) \left( e + \frac{V^2}{2} + g z \right) \end{eqnarray*}

Final Form of the Energy Equation

Now, all of the parts that make up the final form of the energy equation are available in its integral form. To this end, the principle of conservation of energy has been applied to a fluid flowing through a finite control volume that may have heat added or removed, and/or mechanical work is done on the fluid or extracted from it.

Because energy is conserved, then in words, it can be stated that: “The rate of heat added plus the rate of doing work on the flow will be equal to the total rate of change of energy of the flow,” i.e.,

(15)   \begin{equation*} \Delta \overbigdot{Q} + \Delta \overbigdot{W} = \Delta \overbigdot{E} \end{equation*}

which is just the first law of thermodynamics, where \Delta Q is positive when heat is added to the system, and \Delta W is positive when work is done on the system.

(16)   \begin{eqnarray*} \underbrace{\iiint_{\cal{V}} \overbigdot{q} \, \varrho \, d {\cal{V}}}_{\begin{tabular}{c} \footnotesize Net flow rate\\ \footnotesize of heat\\ \footnotesize into C.V. \end{tabular}} & + & \underbrace{\overbigdot{Q}_{\mu}}_{\begin{tabular}{c} \footnotesize Rate of heat\\ \footnotesize added into C.V. \\ \footnotesize from viscous effects \end{tabular}} -\underbrace{ \oiint_{S} (p \, d\vec{S}) \bigcdot \vec{V}}_{\begin{tabular}{c} \footnotesize Work done \\ \footnotesize on the C.S. by \\ \footnotesize pressure forces \end{tabular}} + \underbrace{\iiint_{\cal{V}} ( \varrho \, \vec{f}_b \, d{\cal{V}}) \bigcdot \vec{V}}_{\begin{tabular}{c} \footnotesize Work done \\ \footnotesize on the C.V. by \\ \footnotesize body forces \end{tabular}} \\[12pt] & & + \underbrace{\overbigdot{W}_{\mu}}_{\begin{tabular}{c} \footnotesize Work done \\ \footnotesize on the C.V. \\ \footnotesize by viscous forces \end{tabular}} + \underbrace{\overbigdot{W}_{\rm mech}}_{\begin{tabular}{c} \footnotesize Work done \\ \footnotesize on the C.V. by \\ \footnotesize mechanical sources \end{tabular}} \\[18pt] \hspace*{-200mm} = & \hspace*{-1mm} & \hspace*{-5mm} \underbrace{\frac{\partial}{\partial t} \iiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z\right) d{\cal{V}}}_{\begin{tabular}{c} \footnotesize Time rate of \\ change of energy \\ \footnotesize inside C.V. \end{tabular}} \quad + \quad \underbrace{\oiint_{S} (\varrho \, \vec{V} \bigcdot d\vec{S}) \left( e + \frac{V^2}{2} + g z\right) }_{\begin{tabular}{c} \footnotesize Net flow of energy \\ \footnotesize out of C.V. per \\ \footnotesize unit time \end{tabular}} \end{eqnarray*}

which is mathematically the most general form of the energy equation.  Rewriting it again without the components being identified gives

(17)   \begin{eqnarray*} \iiint_{\cal{V}} \overbigdot{q} \, \varrho \, d {\cal{V}} + \overbigdot{Q}_{\mu} -\oiint_{S} (p \, d\vec{S}) \bigcdot \vec{V} + \iiint_{\cal{V}} ( \varrho \, \vec{f}_b \, d{\cal{V}}) \bigcdot \vec{V} + \overbigdot{W}_{\mu} + \overbigdot{W}_{\rm mech} = & & \nonumber \\ \frac{\partial}{\partial t} \iiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z\right) d{\cal{V}} + \oiint_{S} (\varrho \, \vec{V} \bigcdot d\vec{S}) \left( e + \frac{V^2}{2} + g z\right) \end{eqnarray*}

This latter equation (Eq. 17) now completes the set of three conservation equations, i.e., mass (continuity), momentum, and energy.

Recall for completeness that the continuity equation is

(18)   \begin{equation*} \underbrace{\frac{\partial}{\partial t}\iiint_{\cal{V}} \varrho \, d {\cal{V}}}_{\begin{tabular}{c} \footnotesize Time rate of \\ \footnotesize change of mass \\ \footnotesize inside C.V. \end{tabular}} + \underbrace{ \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S}}_{\begin{tabular}{c} \footnotesize Net mass \\ \footnotesize flow rate \\ \footnotesize out of C.V. \end{tabular}} = 0 \end{equation*}

and the momentum equation is

(19)   \begin{eqnarray*} \vec{F} & = & \underbrace{\iiint_{\cal{V}} \varrho \, \vec{F} _b \, d{\cal{V}}}_{\begin{tabular}{c} \footnotesize Sum of \\ \footnotesize body forces \\ \footnotesize acting on C.V. \end{tabular}} -\underbrace{\oiint_S p \, d\vec{S}}_{\begin{tabular}{c} \footnotesize Sum of \\ \footnotesize pressure forces \\ \footnotesize acting on C.S. \end{tabular}} + \underbrace{ \vec{F}_{\mu}}_{\begin{tabular}{c} \footnotesize Sum of \\ \footnotesize viscous forces \\ \footnotesize acting on C.V. \end{tabular}} \\[12pt] & & = \underbrace{ \frac{\partial}{\partial t}\iiint_{\cal{V}} \varrho \, \vec{V} \, d {\cal{V}}}_{\begin{tabular}{c} \footnotesize Time rate of \\ \footnotesize change of momentum \\ \footnotesize inside C.V. \end{tabular}} + \underbrace{\oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V}}_{\begin{tabular}{c} \footnotesize Net flow of \\ momentum \\ \footnotesize out of C.V. \\ \footnotesize per unit time \end{tabular}} \end{eqnarray*}

In such collective equations, the unknown parameters may include the velocities, pressures, densities, and temperatures, although not all quantities may be unknown or required in any given problem. A fourth equation is also available to round out the set, namely the equation of state p = \varrho R T, which is “handy” because it reduces the number of unknown properties by one.

Simplifications of the Energy Equation

The general form of the energy equation in Eq. 17 may look somewhat forbidding, and it is if written out in its entirety. Nevertheless, as with the other conservation equations, the general form can be written in various simplified forms depending on the assumptions that might be made (and justified!), e.g., steady flow, absence of body forces, no heat added to the system, no mechanical work, no viscous forces, one-dimensional flow, etc. However, because thermodynamic principles are being used here, caution must be applied in using the energy equation to ensure that the needed terms are retained in all potential simplification processes. Remember that in engineering problem solving, any over-simplification of the governing equations without careful justification is likely to meet with disastrous predictive outcomes.

Single Steam System

By way of illustrating the steps that could be used in the simplifications of the energy equation, many practical engineering problems involve fluid systems with just one inlet and one outlet, the mass flow rate through such a system being constant, which are sometimes referred to as single-stream systems. A general depiction of such a single-stream system is shown in the figure below, where some flow comes in at one side of the system and exits on the other. Notice that, in general, there could be a difference in the heights of the inlet and the outlet, so the gravitational potential energy or “head” terms must be retained.

 

The idea of a single-stream device to which heat and mechanical work are added.

For such a single-stream system, the general form of the energy equation can be reduced to the form

(20)   \begin{equation*} \overbigdot{Q} + \overbigdot{W}_{\rm shaft} = \overbigdot{m} \left( (e_2 - e_1) + \frac{V_2^2 - V_1^2}{2} + g ( z_2 - z_1 ) + \left( \frac{p_2}{\varrho_2} - \frac{p_1}{\varrho_1} \right) \right) \end{equation*}

where the subscripts 1 and 2 refer to the inlet and outlet conditions, respectively.

Look carefully at Eq. 17 and consider how Eq. 20 is obtained. As you do that, remember that a mass flow rate or the “\overbigdot{m} term” is always viewed as the flow velocity times the density of the flow times the area through which the flow passes. Notice also the form of the pressure term. The rate of doing work by the pressure forces is equivalent to the product of the pressure times an area times a velocity, so the pressure term in Eq. 20 becomes equivalent to a net pressure contribution times the mass flow rate divided by density.

The left-hand side of the preceding equation represents the energy input, and the right-hand side represents the energy output. The energy input, in this case, comes from the rate of heat transfer to the fluid \overbigdot{Q}, the rate of mechanical work being done on the fluid \overbigdot{W}_{\rm mech} (such as through a shaft driven from outside of the system by a pump), as well as any work done by the pressure forces.

In such a system, the mass flow through the system is conserved (mass inside the system is constant if the flow is steady), so based on per unit mass, which means dividing the terms in the equations through by \overbigdot{m}, then Eq. 20 becomes

(21)   \begin{equation*} q + w_{\rm shaft} = e_2 - e_1 + \frac{V_2^2 - V_1^2}{2} + g ( z_2 - z_1) + \left( \frac{p_2}{\varrho_2} - \frac{p_1}{\varrho_1} \right) \end{equation*}

which is a common form of the steady flow energy balance in thermodynamics. Rearranging this equation gives

(22)   \begin{equation*} w_{\rm shaft} = \frac{V_2^2 - V_1^2}{2} + g ( z_2 - z_1 ) + \left( \frac{p_2}{\varrho_2} - \frac{p_1}{\varrho_1} \right) + ( e_2 - e_1 - q ) \end{equation*}

The latter term in Eq. 22 can be viewed as the sum of the frictional losses, i.e., the value of (e_2 - e_1 - q) > 0 in accordance with the second law of thermodynamics. Such losses can be considered unavailable energy, most likely lost from a system as heat conduction, radiation, or viscosity.

Another way to write this previous equation is just

(23)   \begin{equation*} w_{\rm shaft} = \frac{V_2^2 - V_1^2}{2} + g ( z_2 - z_1 ) + \left( \frac{p_2}{\varrho_2} - \frac{p_1}{\varrho_1} \right) + \mbox{losses} \end{equation*}

and with rearrangement, then

(24)   \begin{equation*} \frac{p_1}{\varrho_1} + \frac{V_1^2}{2} + g z_1 + w_{\rm shaft} = \frac{p_2}{\varrho_2} + \frac{V_2^2}{2} + g z_2 + \mbox{losses} \end{equation*}

which is often called the mechanical energy equation in terms of work per unit mass.

What are the units of Eq. 24? For this equation to be dimensionally homogeneous, then the units of each of the terms must be the same, i.e.,

    \[ \left[ \frac{p}{\varrho} \right] = \frac{M L T^{-2} (L^{-2}) }{M L^{3}} = L^2 T^{-2} \]

and

    \[ \left[ \frac{V^2}{2} \right] = ( L T^{-1})^2 = L^2 T^{-2} \]

and

    \[ \left[ g z \right] = (L T^{-2}) L = L^2 T^{-2} \]

Therefore, the equation has been confirmed to be dimensionally homogeneous with units of L^2 T^{-2}. Remember that work, W, has units of [W] = M L^2 T^{-2}, therefore, units of L^2 T^{-2} represent work per unit mass.

Special Case: No Losses

Now, suppose the flow is ideal with no irreversible processes such as friction, then the total energy must be conserved. Then the energy loss term (e_2 - e_1 - q) \equiv 0, i.e., in that case, the internal energy does not change from frictional effects. Now, an even simpler form of the energy conservation equation is available, i.e.,

(25)   \begin{equation*} \frac{p_1}{\varrho_1} + \frac{V_1^2}{2} + g z_1 + w_{\rm shaft} = \frac{p_2}{\varrho_2} + \frac{V_2^2}{2} + g z_2 \end{equation*}

or in terms of power, so bringing the \overbigdot{m} back, gives

(26)   \begin{equation*} \overbigdot{m} \bigg( \frac{p_1}{\varrho_1} + \frac{V_1^2}{2} + g z_1 \bigg) + \overbigdot{W}_{\rm shaft} = \overbigdot{m} \bigg( \frac{p_2}{\varrho_2} + \frac{V_2^2}{2} + g z_2 \bigg) \end{equation*}

Mechanical Work

Remember that adding or subtracting mechanical work, W_{\rm shaft}, to a flow is common in aerospace thermodynamic systems, such as utilizing a pump, fan, turbine, or propeller, W_{\rm shaft} can be positive or negative. The addition or subtraction of mechanical work to or from the system may appear as a change in the flow’s kinetic energy and/or to the potential energy of the flow and/or to the internal temperature and/or pressure in the system or any combination.

In the case of a basic pump, which adds energy and does work on the fluid system, then

(27)   \begin{equation*} \overbigdot{W}_{\rm shaft} = \overbigdot{W}_{p} \end{equation*}

and for the turbine, which takes energy and work from the fluid system to produce mechanical work and power output at a shaft, then

(28)   \begin{equation*} -\overbigdot{W}_{\rm shaft} = \overbigdot{W}_{t} \end{equation*}

Therefore, another way of writing the energy equation is

(29)   \begin{equation*} \underbrace{ \overbigdot{m} \bigg( \frac{p_1}{\varrho_1} + \frac{V_1^2}{2} + g z_1 \bigg) }_{\begin{tabular}{c} \footnotesize Rate of energy\\ \footnotesize entering C.V. \end{tabular}}+ \underbrace{ \overbigdot{W}_{p}}_{\begin{tabular}{c} \footnotesize Rate of \\ \footnotesize energy \\ \footnotesize added \\ \footnotesize by pump \end{tabular}}= \underbrace{ \overbigdot{m} \bigg( \frac{p_2}{\varrho_2} + \frac{V_2^2}{2} + g z_2 \bigg) }_{\begin{tabular}{c} \footnotesize Rate of energy\\ \footnotesize leaving C.V. \end{tabular}} + \underbrace{ \overbigdot{W}_{t}}_{\begin{tabular}{c} \footnotesize Rate of \\ \footnotesize energy \\ \footnotesize subtracted \\ \footnotesize by turbine \end{tabular}} \end{equation*}

Finally, if the fluid is assumed to be incompressible, i.e., \varrho_1 = \varrho_2 = \varrho, then

(30)   \begin{equation*} \overbigdot{m} \bigg( \frac{p_1}{\varrho} + \frac{V_1^2}{2} + g z_1 \bigg) + \overbigdot{W}_{p} = \overbigdot{m} \bigg( \frac{p_2}{\varrho} + \frac{V_2^2}{2} + g z_2 \bigg) + \overbigdot{W}_{t} \end{equation*}

or in terms of per unit mass (again, divide by \overbigdot{m}), then

(31)   \begin{equation*} \frac{p_1}{\varrho} + \frac{V_1^2}{2} + g z_1 + {w}_{p} = \frac{p_2}{\varrho} + \frac{V_2^2}{2} + g z_2 + {w}_{t} \end{equation*}

which is called the extended Bernoulli equation.

Energy Equation in Terms of “Head”

Civil and hydraulic engineers, who deal primarily with liquids rather than gases, often express the energy equation in terms of pressure head and specific weight. The pressure head or the static pressure head, h, is the height of a liquid column corresponding to a particular pressure value so that it will have units of length, i.e., meters or feet. After doing some problems with the energy equation, it will be apparent that using a pressure head is a convenient concept.

Specific Weight

The specific weight is the weight of a unit volume of a fluid. The symbol \gamma is often used for specific weight, but this choice causes a symbol conflict with aerospace and mechanical engineers who use \gamma as the ratio of specific heats. Therefore, specific weight can be denoted by the symbol w, i.e.,

(32)   \begin{equation*} w = \frac{\mbox{Weight~of~fluid}}{\mbox{Volume~of~fluid}} = \frac{ \varrho \, {\cal{V}} \, g}{{\cal{V}}} = \varrho \, g \end{equation*}

or by rearrangement, then

(33)   \begin{equation*} \varrho = \frac{w}{g} \end{equation*}

Using the specific weight, which can be obtained by dividing through Eq. 36 by g, the incompressible energy equation in terms of the pressure head can be written as

(34)   \begin{equation*} \frac{p_1}{\varrho \, g} + \frac{V_1^2}{2g} + z_1 + {h}_{p} = \frac{p_2}{\varrho \, g} + \frac{V_2^2}{2g} + z_2 + {h}_{t} or \end{equation*}

(35)   \begin{equation*} \frac{p_1}{w} + \frac{V_1^2}{2g} + z_1 + {h}_{p} = \frac{p_2}{w} + \frac{V_2^2}{2g} + z_2 + {h}_{t} \end{equation*}

where the units of all terms are now length, so m or ft. If frictional losses are included, which can also be expressed in terms of head, i.e., {h}_{l}, then

(36)   \begin{equation*} \frac{p_1}{w} + \frac{V_1^2}{2g} + z_1 + {h}_{p} = \frac{p_2}{w} + \frac{V_2^2}{2g} + z_2 + {h}_{t} + {h}_{l} \end{equation*}

Pressure Head of a Pump or Turbine

The power, P, of a pump or turbine can be expressed as

(37)   \begin{equation*} P = \overbigdot{m} \, g \, h \end{equation*}

The mass flow rate, \overbigdot{m}, is

(38)   \begin{equation*} \overbigdot{m} = \varrho \, \overbigdot{{\cal{V}}} \end{equation*}

where \overbigdot{{\cal{V}}} is the volume flow rate. Therefore,

(39)   \begin{equation*} P = w \, \overbigdot{{\cal{V}}} \, h \end{equation*}

Of course, no pump or turbine can be 100% efficient, so if the efficiency is \eta, then the power is

(40)   \begin{equation*} P = \frac{w \, \overbigdot{{\cal{V}}} \, h}{\eta} \end{equation*}

Therefore, in the case of a pump (energy in) of power P_p, then

(41)   \begin{equation*} h_p = \frac{ \eta_p \, P_p}{ w \, \overbigdot{{\cal{V}}}} \end{equation*}

In the case of a turbine (energy out) of power P_t, then

(42)   \begin{equation*} h_t = \frac{ \eta_t \, P_t}{ w \, \overbigdot{{\cal{V}}}} \end{equation*}

Caution: Do not confuse the pressure head for a pump, h_p, with horsepower units, given the symbol “hp.”

Worked Example #1 – Using the energy equation to calculate pumping power

Use conservation of energy principles to calculate the power of a hydraulic pump that must deliver the fluid (oil) from one location to another. The pump is 75% efficient in converting mechanical input work to pressure. The inlet pressure, p_1, is 100 kPa, and the outlet pressure, p_2, is 500 kPa. The diameter of the inlet pipe, d_1, is 10 cm, and the outlet pipe, d_2, is 5 cm. The flow rate, \overbigdot{{\cal{V}}}, is 60 m3/hr. The height difference, z_2 - z_1, is 3.2 m. The density of the hydraulic fluid is 850 kg/m3. Assume the internal fluid losses are equivalent to 1.4 m of static pressure head.

First, it is necessary to find the inlet and outlet velocities, V_1 and V_2, respectively, using

    \[ V_1 = \frac{\overbigdot{{\cal{V}}}}{A_1} = \frac{4 \overbigdot{{\cal{V}}}}{\pi d_1^2} = \frac{4 \times 60.0/3,600}{\pi \times 0.1^2} =2.12~\mbox{m/s} \]

and for V_2, then

    \[ V_1 = \frac{\overbigdot{{\cal{V}}}}{A_2} = \frac{4 \overbigdot{{\cal{V}}}}{\pi d_2^2} = \frac{4 \times 60.0/3,600}{\pi \times 0.05^2} =8.49~\mbox{m/s} \]

The relevant form of the energy equation is

    \[ \frac{p_1}{\rho \, g} + \frac{V_1^2}{2g} + z_1 + {h}_{p} = \frac{p_2}{\rho \, g} + \frac{V_2^2}{2g} + z_2 + h_l \]

so solving for h_p gives

    \[ h_{p} = \frac{p_2 - p_1}{\rho \, g} + \frac{V_2^2 - V_1^2}{2g} + (z_2 - z_1) + h_l \]

Inserting the known values gives

    \[ h_{p} = \frac{(500.0 - 100.0) \times 10^3}{850.0 \times 9.81} + \frac{8.49^2 - 2.12^2}{2.0 \times 9.81} + 3.2 + 1.4 = 56.15~\mbox{m} \]

The power required from the pump is

    \[ P_p = \frac{ \varrho \, g \, \overbigdot{{\cal{V}}} \, h_p}{\eta_p} \]

and inserting the numerical values gives

    \[ P_p = \frac{ 850.0 \times 9.81 \times (60.0/3,600) \times 56.15}{0.75} = 10.41 ~\mbox{kW} \]

Bernoulli’s Equation

By assuming incompressible flow and that no mechanical work is introduced into or taken out of the fluid system, then {w}_{p} = {w}_{t} = 0 and so Eq. 36 becomes

(43)   \begin{equation*} \frac{p_1}{\varrho} + \frac{V_1^2}{2} + g z_1 = \frac{p_2}{\varrho} + \frac{V_2^2}{2} + g z_2 \end{equation*}

This latter equation can then be rearranged into the form

(44)   \begin{equation*} p_1 + \frac{1}{2} \varrho V_1^2 + \varrho g z_1 = p_2 + \frac{1}{2} \varrho V_2^2 + \varrho g z_2 \end{equation*}

or simply that

(45)   \begin{equation*} p + \frac{1}{2} \varrho V^2 + \varrho g z = \mbox{constant} \end{equation*}

which is known as the Bernoulli equation or Bernoulli’s principle after Daniel Bernoulli.

Notice that the Bernoulli equation has units of pressure. It will be apparent that the Bernoulli equation is a statement of energy conservation in that a fluid exchanges its specific kinetic energy for pressure, either static or potential, the specific kinetic energy being the kinetic energy per unit volume.

A more general form of the Bernoulli equation, which often appears in many sources, is to leave it in an unintegrated form, i.e.

(46)   \begin{equation*} \frac{dp}{\varrho} + \frac{1}{2} d( V^2 ) + g z = 0 \end{equation*}

After integration then

(47)   \begin{equation*} \int \frac{dp}{\varrho} + \frac{1}{2} V^2 + g z = \mbox{constant} \end{equation*}

Assuming incompressible flow gives

(48)   \begin{equation*} \frac{p}{\varrho} + \frac{1}{2} V^2 + g z = \mbox{constant} \end{equation*}

or

(49)   \begin{equation*} p + \frac{1}{2} \varrho V^2 + \varrho g z = \mbox{constant} \end{equation*}

as has been written down previously.

The Bernoulli equation is one of the most famous fluid mechanics equations and is used to solve many practical problems. It has been derived here as a particular degenerate case of the general energy equation for a steady, inviscid, incompressible flow. Still, it can also be derived in several other ways. Remember that it is not an energy equation per se because it has units of pressure, so it is often referred to as a surrogate of the energy equation.

Alternative Derivation of Bernoulli’s Equation

In terms of another physically intuitive derivation of the Bernoulli equation, consider the figure below, which represents a streamtube flow of an ideal fluid that is steady, incompressible, and inviscid. At the inlet to the streamtube, the cross-sectional area is A_1 with flow velocity V_1 and at a height z_1 from a reference level. At the outlet, the area is A_2, and the velocity is V_2 at height Z_2. In a given time, a mass of fluid dm can be considered to have moved in the streamtube from the inlet to the outlet.

Streamtube flow model used for an alternative derivation of the Bernoulli equation.

Let p_1 be the pressure acting uniformly at the inlet at station 1. Because pressure acts inward, the pressure creates a force pushing the fluid downstream toward the outlet. The work done by the pressure force here will be

(50)   \begin{equation*} W_1 = \left( p_1 \, A_1 \right) dx_1 \end{equation*}

remembering that work is equal to force times distance. Let p_2 be the pressure acting uniformly at the outlet at station 2. In this case, the work done by the pressure force will be

(51)   \begin{equation*} W_2 = -\left( p_2 \, A_2 \right) dx_2 \end{equation*}

the minus sign indicates that the pressure force here acts upstream opposite to the fluid flow. In addition, there will be work done under gravity, which will be

(52)   \begin{equation*} W_g = - dm \, g \left( z_2 - z_1 \right) = - \varrho \, d{\cal{V}} \, g \left( z_2 - z_1 \right) \end{equation*}

where the mass dm = \varrho \, d{\cal{V}}, and the minus sign means that work must be put into the fluid system to increase its potential energy.  Therefore, the total external work is

(53)   \begin{equation*} W_1 + W_2 + W_g = p_1 \, A_1 \, dx_1 - p_2 \, A_2 \, dx_2 - \varrho \, d{\cal{V}} \, g \left( z_2 - z_1 \right) \end{equation*}

The kinetic energy of the flow must now be considered. The change in kinetic energy of the fluid as it moves through the streamtube is

(54)   \begin{equation*} \Delta KE = \frac{1}{2} dm V_2^2 - \frac{1}{2} dm V_1^2 = \frac{1}{2} \, \varrho \, d {\cal{V}} \, V_2^2 - \frac{1}{2} \, \varrho \, d{\cal{V}} \, V_1^2 \end{equation*}

The application of the principle of conservation of energy requires that

(55)   \begin{equation*} W_1 + W_2 + W_g = \Delta KE \end{equation*}

so that

(56)   \begin{equation*} p_1 \, A_1 \, dx_1 - p_2 \, A_2 \, dx_2 - \varrho \, d{\cal{V}} \, g \left( z_2 - z_1 \right) = \frac{1}{2} \, \, \varrho \, d{\cal{V}} \, V_2^2 - \frac{1}{2} \, \varrho \, d{\cal{V}} \, V_1^2 \end{equation*}

Notice also that conservation of mass (the continuity equation) requires that

(57)   \begin{equation*} \varrho \, d{\cal{V}} = \varrho \left( A_1 dx_1 \right) = \varrho \left( A_2 dx_2 \right) \end{equation*}

or just

(58)   \begin{equation*} A_1 \, dx_1 = \, A_2 \, dx_2 \end{equation*}

so

(59)   \begin{equation*} \left( p_1 - p_2 \right) d{\cal{V}} - \varrho \, d{\cal{V}} \, g \, \left( z_2 - z_1 \right) = \frac{1}{2} \, \varrho \, d{\cal{V}} \, \left( V_2^2 - V_1^2 \right) \end{equation*}

Cancelling out the volume d{\cal{V}} and rearranging this latter equation gives

(60)   \begin{equation*} p_1 + \frac{1}{2} \, \varrho V_1^2 + \varrho g z_1 = p_2 + \frac{1}{2} \, \varrho V_2^2 + \varrho g z_2 \end{equation*}

or

(61)   \begin{equation*} p + \frac{1}{2} \, \varrho V^2 + \varrho g z = \mbox{constant} \end{equation*}

which, again, is the Bernoulli equation.

Caution: Remember that in the derivation of the Bernoulli equation, the flow has been assumed to be steady and incompressible with negligible frictional losses (i.e., an ideal fluid) and where no mechanical work is added or subtracted. While the Bernoulli equation is found to have many practical applications, it should be remembered that it has been derived based on these prior assumptions, so its practical use requires considerable justification and caution in actual application.

Worked Example #2 – Pressure change in a contraction

Air flows at low speed through a pipe with a volume flow rate of 0.135 m^3/s. The pipe has an inlet section diameter of 21 cm and an outlet section diameter of 9 cm. A water manometer measures the pressure difference between the inlet and outlet sections. Assuming no frictional losses, determine the differential height \Delta h shown on the manometer. Take the density of air to be 1.21 kg/m^3 and the density of water to be 1,000 kg/m^3. Make any assumptions you feel are justified.

From the information, assuming a one-dimensional, steady, incompressible, inviscid flow seems reasonable. The flow rates, velocities, and Mach numbers are low enough that compressibility effects can be neglected. Let the inlet be condition 1 and the outlet condition 2. The continuity equation relates the inlet and outlet conditions, i.e.,

    \[ \overbigdot{m} = \varrho A_1 V_1 = \varrho A_2 V_2 = \mbox{constant} \]

Or because the flow is assumed incompressible, then just

    \[ \overbigdot{{\cal{V}}} = A_1 V_1 = A_2 V_2 = \mbox{constant} = 0.135~\mbox{m$^3$/s} \]

It follows that

    \[ V_1 = \frac{\overbigdot{{\cal{V}}} }{A_1} \]

and

    \[ V_2 = \frac{\overbigdot{{\cal{V}}} }{A_2} \]

so it is possible to calculate A_1, i.e.,

    \[ A_1 = \frac{\pi d_1^2}{4} = \frac{\pi \, 0.21^2}{4} = 0.0346~\mbox{m$^2$} \]

and also A_2, i.e.,

    \[ A_2 = \frac{\pi d_2^2}{4} = \frac{\pi \, 0.09^2}{4} = 0.00636~\mbox{m$^2$} \]

so that

    \[ V_1 = \frac{\overbigdot{{\cal{V}}} }{A_1} = \frac{0.135}{0.0346} = 3.90~\mbox{m/s} \]

and

    \[ V_2 = \frac{\overbigdot{{\cal{V}}} }{A_2} = \frac{0.135}{0.00636} = 21.22~\mbox{m/s} \]

which are much lower than a Mach number of 0.3, so the assumption of incompressible flow is justified.

The pressure difference between points 1 and 2 is needed, which requires some form of the energy equation. Using the Bernoulli equation is justified because the information given is for an incompressible, frictionless flow. The Bernoulli equation is

    \[ p_1 + \frac{1}{2} \varrho V_1^2 = p_2 + \frac{1}{2} \varrho V_2^2 \]

so

    \[ p_1 - p_2 = \frac{1}{2} \varrho V_2^2 - \frac{1}{2} \varrho V_1^2 = \frac{1}{2} \varrho \left( V_2^2 - V_1^2 \right) \]

Inserting the values for \varrho, V_1 and V_2 gives

    \[ p_1 - p_2 = \frac{1}{2} \varrho \left( V_2^2 - V_1^2 \right) = \frac{1}{2} \times 1.21 \left( 21.22^2 - 3.9^2 \right) = 263.25~\mbox{Pa} \]

In terms of h, which is the differential height on the manometer, then

    \[ p_1 - p_2 = \varrho_{\rm H_2 0} \, g \, h = 263.25~\mbox{Pa} \]

so solving for h, gives

    \[ h = \frac{p_1 - p_2 }{\varrho_{\rm H_2 0} \, g} = \frac{263.25}{1,000 \times 9.81} = 0.0268~\mbox{m} = 2.68~\mbox{cm} \]

Other Compressible Forms of the Bernoulli Equation

The unintegrated form of the Bernoulli equation is

(62)   \begin{equation*} \int \frac{dp}{\varrho} + \frac{1}{2} V^2 + g z = \mbox{constant} \end{equation*}

Recall that for an incompressible flow, the first term is

(63)   \begin{equation*} \int \frac{dp}{\varrho} = \frac{p}{\rho} \end{equation*}

so the classic Bernoulli equation is recovered, i.e.,

(64)   \begin{equation*} p + \frac{1}{2} \rho V^2 + \rho g z = \mbox{constant} \end{equation*}

Isothermal Process

For an isothermal process, then T = constant, so from the equation of state, then

(65)   \begin{equation*} \frac{p}{\varrho} = R \, T = \mbox{constant} = C_1 \end{equation*}

or

(66)   \begin{equation*} \varrho = \frac{p}{C_1 } \end{equation*}

Therefore,

(67)   \begin{equation*} \int \frac{dp}{\varrho} = C_1 \ln p = \frac{p}{\varrho} \ln p \end{equation*}

and so the Bernoulli equation for an isothermal flow is

(68)   \begin{equation*} \frac{p}{\varrho} \ln p + \frac{1}{2} V^2 + g z = \mbox{constant} \end{equation*}

Adiabatic Process

Another case of a compressible flow is an adiabatic process characterized by the relationship

(69)   \begin{equation*} \frac{p}{\varrho^{\gamma}} = C_2 = \mbox{constant} \end{equation*}

where \gamma is the ratio of specific heats. Solving for the density \varrho gives

(70)   \begin{equation*} \varrho = C_2^{(-1/\gamma)} p^{1/\gamma} \end{equation*}

As previously derived, the unintegrated form of the Bernoulli equation can be written as

(71)   \begin{equation*} \int \frac{dp}{\varrho} + \frac{1}{2} V^2 + g z = \mbox{constant} \end{equation*}

Substituting for \varrho into the first term gives

(72)   \begin{equation*} \int \frac{dp}{\varrho} = \int C_2^{(1/\gamma)} p^{(-1/\gamma)} dp = \left( \frac{\gamma}{\gamma - 1} \right) \frac{p}{\varrho} \end{equation*}

Therefore, another form of the Bernoulli equation for the steady, adiabatic, compressible flow of a gas becomes

(73)   \begin{equation*} \left( \frac{\gamma}{\gamma - 1} \right) \frac{p}{\varrho} + \frac{1}{2} V^2 + g z = \mbox{constant} \end{equation*}

Summary & Closure

Applying the principle of energy conservation to fluid flow results in a rather formidable-looking equation in its more general form. The energy equation is developed from thermodynamic principles, i.e., the first law of thermodynamics, and uses the concepts of heat, work, and power. The types of energy are internal, potential, and kinetic energy, all of which are involved in most fluid problems. The energy equation is a power equation based on its units, i.e., the rate of doing work. When thermodynamic principles are involved in fluid flow problems, the equation of state is also helpful in establishing the relationships between the known and unknown quantities.

The most common application of the energy equation is to so-called single-stream systems, in which a certain amount of fluid energy comes into a system where work is added or extracted. Then, the energy can come out of the system in another form. Further simplifying the energy equation to incompressible, inviscid flows without energy addition leads to the Bernoulli equation, which can be used successfully in many practical problems to relate pressures and flow velocities. The Bernoulli equation is also a statement of energy conservation, i.e., fluid exchanges its specific kinetic energy for static or potential pressure. Nevertheless, the Bernoulli equation must be used carefully and consistently applied within the assumptions and limitations of its derivation.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • The energy equation is often called a redundant equation for analyzing incompressible, inviscid flows. Why?
  • What are the three main assumptions used in deriving the Bernoulli equation?
  • The flow through the turbine blades of a jet engine can be modeled using the incompressible form of the Bernoulli equation. True or false? Explain.
  • The Bernoulli equation can be used to analyze the flow through a propeller operating at low flow speeds. Explain how to do that.

Additional Online Resources

To learn more about the energy equation, as well as the Bernoulli equation and its uses, check out some of these online resources:

  • A good video on the first law of thermodynamics and internal energy.
  • Video sequence from Dr. Biddle’s YouTube channel on fluid mechanics.
  • A video lecture describing the derivation of the energy equation for a fluid.
  • Video on conservation of energy in control volume form.
  • This a great video on better understanding the Bernoulli equation.
  • The use of graphics in this video reinforces the meaning of the Bernoulli equation.
  • Another good video describing the applications of the Bernoulli equation.

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n18