Energy Equation in Flow Systems

When energy crosses the boundary of a system, the energy in the system changes by an equal amount. Thus a decrease in the energy of a system equals the increase in energy of the surroundings and vice versa.

Energy can cross a system boundary in the form of heat transfer (to or from the system), in the form of work (done by or on the system), or by mass entering or leaving. A process during which no heat is transferred to or from a system is known as an adiabatic process. The law of conservation of energy states that for a system of particles

E₂ – E₁ = Q͂ - Wʹ

Or

dE = d Q͂ - dWʹ

Where:

• E2 – E1: is the change in energy experienced by the system
• Q͂: is the heat added to the system
• Wʹ: is work done by the system

The total energy is a property of the system and includes internal (U), kinetic (KE), and potential energies (PE):

E = U + KE + PE

If we express this on a per-unit-mass basis, we obtain:

Heat and work are not properties of the system, but they represent energy transfers moving across the system boundaries in specific forms.

Remember the general conservation expression:

If we substitute, N = E and n = e, we obtain:

Equation 1 is generally known as the first law of thermodynamics and is a statement of the conservation of energy.

The Wʹ term consists of all forms of work crossing the boundary. This includes shaft work, electric and magnetic work, viscous shear work, and flow work. The last of these forms represents work done by the fluid system in pushing mass into or out of the control volume. Thus:

where dW_f /dt is flow work per unit time.

To develop an expression for dW_f/dt, we can consider an element of mass dm leaving a control volume through an area dA, as shown below:

The fluid in the control volume must work against the external pressure p to move dm out. Since work = force x distance, we have:

Similarly, for an element of mass entering, work is being done on the system. Therefore, we get:

Combining the previous two equations, we can obtain the flow work as:

If we substitute Equations 3 and 2 into Equation 1 we get:

Combining the integral terms results in:

Enthalpy is a property defined as h = u + p/ρ; the conservation of energy equation now becomes:

For a steady, one-dimensional flow, the energy equation becomes:

For many incompressible flows, Q͂ is usually assumed to be zero, and changes in internal energy are frequently negligible. Thus equation 5 can be simplified into:

For compressible flows, however, the change in internal energy is significant and cannot be neglected. 

The Heat Diffusion Equation

It is important to know the temperature distribution as it allows us to see how temperature varies with position inside a medium. Once the distribution is known, Fourier’s law can be used to calculate the conduction heat flux at any point in the medium or on its surface or any other important quantities of interest.

In a solid, temperature distribution allows study of structural integrity by determining thermal stress, expansions and deflections. Temperature distribution can also help to optimize the thickness of an insulating materials, determine the compatibility of special coatings or adhesives used within the material.

In order to determine the temperature distribution, we can follow the same methodology used in the chapter on Relationship to the First Law of Thermodynamics (see here) of applying energy conservation. In this case, we:

• define a differential control volume,
• identify the relevant energy transfer processes
• introduce the appropriate rate equations

All this results in a differential equation whose solution for a set of boundary conditions, provides the temperature distribution in the medium.

Let us consider a homogeneous medium within which there is no advection (bulk motion) and the temperature distribution T(x, y, z) is expressed in Cartesian coordinates. We then define an infinitesimally small differential control volume dx·dy·dz as shown below:

Formulate the first law at an instant of time, followed by considering the energy processes relevant to this control volume. If there is no motion or with uniform motion, there are no changes in the mechanical energy and no work is being done on the system. Thus, only thermal forms of energy need to be considered. If there are temperature gradients, heat transfer through conduction will occur across each of the control surfaces. The conduction heat rates perpendicular to each control surfaces at the x-, y- and z-coordinate locations are expressed by the terms q_x, q_y, and q_z, respectively. Using Taylor series expansion and neglecting higher-order terms, we can express the conduction heat rates at the opposite surfaces as:

Equation 1a, states that the x-component of the heat transfer rate at x + dx is equal to the value of this component at x plus the amount by which is changes with respect to x times dx.

Within the medium there may also be an energy source term associated with the rate of thermal energy generation (E_g). This term is represented as:

E_g = q dx dy dz                 (2)

where q is the rate at which energy is generated per unit volume of the medium (W/m³).

Additionally, changes may occur in the amount of the internal thermal energy stored by the material in the control volume. If the material is not experiencing a change in phase, latent energy effects are not relevant, and the energy storage term may be expressed as:

E_st = ρC_p(∂T / ∂t) dx dy dz                    (3)

where ρC_p(∂T / ∂t) is the time rate of change of the sensible (thermal) energy of the medium per unit volume

The terms E_g and E_st represent different physical processes. The energy generation term (E_g) is an indicator of some energy conversion process involving thermal energy on one hand and some other form of energy like chemical, electrical, or nuclear, on the other. The term is positive (a source) if thermal energy is being generated in the material at the expense of some other energy form; it is negative (a sink) if thermal energy is being consumed. In contrast, the energy storage term (E_st) refers to the rate of change of thermal energy stored by the matter.

On a rate basis, the general form of the conservation of energy requirement is:

E_in + E_g – E_out  = E_st

Hence, recognizing that the conduction rates constitute the energy inflow E_in and outflow E_out, and substituting Equations 2 and 3, we obtain:

Substituting from Equations 1:

The conduction heat rates in an isotropic material may be evaluated using Fourier’s law:

where each heat flux component of Equation 4 has been multiplied by the appropriate control surface (differential) area to obtain the heat transfer rate. Substituting Equations 6 into Equation 5 and dividing out the dimensions of the control volume (dx dy dz), we obtain: 

Equation 7 is the general Cartesian coordinates form of the heat diffusion equation. This equation is referred to as the heat equation and is the basic tool for heat conduction analysis and its solution helps us obtain the temperature distribution T(x, y, z) as a function of time. The heat equation describes the conservation of energy.  In simple words, Equation 7 means that at any point in the medium, the net rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must be equal to the rate of change of thermal energy stored within the volume.

In equation 7, the term ∂(k∂T/∂x)/∂x is related to the net conduction heat flux into the control volume for the x-coordinate direction. That is, multiplying by dx we get:

The flux expressions in the y- and z- directions are similar to equation 8.

It is often possible to simplify Equation 7. For example, if the thermal conductivity is constant, the heat equation is:

where α = k/ρc_p is the thermal diffusivity.

Additional simplifications of the general form of the heat equation are also possible. For example, under steady-state conditions, there can be no change in the amount of energy storage; hence Equation 7 reduces to:

If the heat transfer is one-dimensional (e.g., in the x-direction) and there is no energy generation, Equation 10 reduces to:

This shows that, under steady-state, one-dimensional conditions with no energy generation, the heat flux is a constant in the direction of transfer.

The heat equation may also be expressed in cylindrical and spherical coordinates.

i) Cylindrical Coordinates

The differential control volume for cylindrical coordinates is as shown below:

When we express Equation 3 from The Conduction Rate Equation blog entry (see here) in cylindrical coordinates, the general form of the heat flux vector and hence of Fourier’s Law is:

Where equations 13 represent the heat flux components in the radial (r), circumferential (ϕ), and axial directions (z), respectively.

Applying an energy balance to the cylindrical coordinate’s differential control volume, the following general form of the heat equation is obtained:

ii) Spherical Coordinates

The differential control volume for cylindrical coordinates is as shown below:

The general form of the heat flux vector and Fourier’s law in spherical coordinates is:

Equations 16 are the heat flux components in the radial (r), polar (θ), and azimuthal directions (ϕ), respectively.

Applying an energy balance to the spherical coordinate’s differential control volume, the following general form of the heat equation is obtained:

Momentum Equation

From the previous chapter, we saw that the continuity equation is a conservation of mass equation with which mass transfers across boundaries and mass storage within control volumes can be accounted for.

Now we can derive a conservation of linear and angular momentum equations using the same technique.

Linear Momentum Equation

Using the general conservation equation below:

(S = System, CV = Control volume, CS = Control surface)

Let N be momentum mV, and n, which is N per unit mass, becomes V. Thus, the general conservation equation becomes:

From Newton’s law for the system, we have:

Combining these equations with Equation 1, we get a conservation of linear momentum equation for the three principal directions:

Two velocity terms appear in the integrals of the last terms on the right-hand sides of equations 2a, 2b and 2c. One of the velocities refers to a principal flow direction; for example, V_x. The other velocity, V_n, is normal to the control surface where mass crosses the boundary. These two velocities are not always equal. Because force and velocity have magnitude and direction, they are vectors.

Thus, Equation 2 can be written in vector form as:

This equation is a conservation of linear momentum equation.

As required by Newton’s law, the term ΣF represents all forces applied externally to the control volume. These include forces due to gravity, electric and magnetic fields, surface tension effects, pressure forces, and viscous forces (friction).

The first term on the right-hand side of equation 3 represents the rate of storage of linear momentum in the control volume. The last term is a net rate out (out minus in) of linear momentum from the control volume. For steady one-dimensional flow, Equation 3 becomes, for any direction i,

For one fluid stream entering and one leaving the control volume, we have the following for one-dimensional flow:

The momentum equation can be used to set up a general equation for frictional effects existing within a pipe, for a certain type of phenomenon in open channel flow, and for other kinds of applications-oriented problems.

Angular Momentum Equation

For some fluid mechanics problems, it is essential to be able to evaluate moments exerted by moving fluid volumes especially for rotating machinery like turbines and pumps. The equation applicable to these instances is known as angular momentum equation and is derived using the linear momentum equation.

Let us consider a control volume located in the xy plane as shown below.

This control volume is located a distance, r, from the origin; and as fluid passes through the control volume, a force is exerted. The force can be resolved into two components: one normal and one tangential to r. The torque exerted by the force equals the product of force and moment arm. In differential form, we have:

where: dT₀ is the differential torque exerted about the origin

            dF_t is the tangential component of the differential force perpendicular to r

Because the force is regarded as being caused by fluid motion through the control volume, we can use the linear momentum equation (Equation 2) applied in the tangential direction to obtain the following:

The differential torque is then,

Integrating the above expression over all control volumes that contribute to the total torque, we get:

The left-hand side of equation 5 represents the sum of all externally applied torques on the control volume. On the right-hand side, the first term represents a storage of angular momentum in the control volume; the last term is the net rate out (out minus in) of angular momentum from the control volume.

As seen in xy graph above, V sin θ = V_t. Equation 5 can then be expressed as:

If we use vector notation, the cross product is defined as:

r x V = rV sinθ

Now we can express equation 6 in vector form:

Equation 7 is applicable to both 2-dimensional and 3-dimensional cases. 

THE CONTINUITY EQUATION

The continuity equation is a statement of conservation of mass (covered in this blog entry as Equation 4.)

The flow quantity N becomes m, and n = m/m = 1. Thus, the general conservation equation becomes:

For a system of particles, since mass is neither created nor destroyed mass remains constant. Thus:

This means that the mass entering the control volume is equal to the summation of the mass stored and the mass leaving. For a steady flow with no storage, the mass entering is equal to the mass leaving. Thus, for a steady flow, equation 1 is expressed as:

For incompressible fluids (liquids), ρ is a constant, and the following equation is used for steady incompressible flow:

From Equation 3 we learn that:

• the volume of fluid entering the control volume per unit time is equal to the amount leaving.
• a velocity distribution normal to the control surface must be integrated over the cross-sectional area at inlets and exits to the control volume.

However, for many flow situations, a velocity distribution may not be known or derivable. Thus, in such cases it is more convenient to use average velocity at a cross section that is independent of area – assuming that it is 1-dimensional flow or at least flow that can be estimated as 1-dimensional. Therefore, by definition:

Where V = average velocity. The product AV = Q is known as the volume flow rate with dimensions of L³/T (m³/s or ft³/s). Also used in fluid mechanics is the term ṁ, defined as the mass flow rate, where ṁ = ρQ = ρAV. The mass flow rate has dimensions of M/T (kg/s or slug/s).

To understand average velocity, let us consider a flow in the pipe below with a velocity distribution, V_z:

where R is the pipe radius. The V_z expression or instantaneous velocity, the gives the velocity at any radial location within the pipe. At a control surface normal to the axial direction, V_z = V_n and the average velocity is:

dA = r dr dθ is used in the integral since we are dealing with cylindrical coordinates.

This result means that we can replace the velocity distribution with an average velocity so that the volume flow rate remains unchanged as shown in the velocity distribution figure above. In many practical situations, the volume flow rate and area are known but the velocity distribution isn’t but average velocity can be determined.

When we use the average velocity, the continuity equation becomes:

For steady flow systems where no mass is stored,