Energy Conservation Equation

shayan pakravan

The energy equation, Navier-Stokes equations, and continuity equation are the most important equations in Fluid Mechanics CFD simulations. The energy equation can vary significantly depending on the type of problem and the phenomena involved (such as phase changes or electromagnetism), potentially becoming very complex. Solving, discretizing, and coupling this equation with other equations can be challenging, but ANSYS Fluent manages these tasks effectively. In this article, we will also examine Bernoulli’s equation, which is one of the fundamental equations in fluid mechanics.

Contents

Energy Equation in Fluid Mechanics

In fluid mechanics 3 energy models are usually considered in a system:

$e=e_{internal}+e_{kinetic}+e_{potential}$

Other forms of energy, such as those from chemical reactions, nuclear reactions, and electrostatic or magnetic field effects, may be considered depending on the need. The energy terms of the previous equation, respectively, in units of J/kg are:

$e=u + \frac{1}{2} V^2 +gz$

where u is the internal energy [J/kg], V is the velocity [m/s], g represents the gravitational acceleration [m/s²], and z implies the height [m]. The first law of thermodynamics for a control volume is:

$\frac{dQ}{dt}-\frac{dW}{dt}=\frac{d}{dt}(\int_{CV} e \rho dv)+\int_{CS}e \rho V_n dA$

CS and CV mean control surface and control volume, respectively. dv and dA are the volume and surface elements, respectively. V_n is the velocity component perpendicular to dA, which is positive if it is towards the inside of the control volume and negative if it is towards the outside. Q is the rate of heat exchange between the system and the environment, and W is the exchange of work between the system and the environment. Using for convenience the overdot to denote the time derivative, we divide the work term into three parts:

$\dot{W}=\dot{W}_{shaft}+\dot{W}_{press}+\dot{W}_{viscous \ stresses}=\dot{W}_s+\dot{W}_p+\dot{W}_v$

The work of the shaft is transferred to the system by a mechanical part such as a pump impeller, fan blade, or piston protruding through the control urface into the control volume. The total pressure work is:

$\dot{W}_p=\int_{CS}pV_ndA$

The shear work resulting from viscous stresses occurs at the control surface and is composed of the product of each viscous stress (one normal and two tangential) and the corresponding velocity component.

$\dot{W}_v=-\int_{CS}\tau . \vec{ V } dA$

If the walls of the control surface are solid and the no-slip condition is applied, the speed at the surface is zero, and the shear work is zero. At inlets and outlets, fluid flow is approximately perpendicular to the differential area element dA. Consequently, the only viscous work term arises from the normal stress. Viscous normal stresses are typically negligible except in rare instances, such as within shock waves. Thus, it is standard practice to disregard viscous work at a control volume’s inlets and outlets.

Finally, according to all the mentioned parameters, the energy equation in fluid mechanics is:

$\dot{Q}-\dot{W}_s-\dot{W}_v=\frac{\partial}{\partial t}\int_{CV}\left( u+\frac{1}{2}V^2+gz \right) \rho dv + \int_{CS}\left( u+\frac{1}{2}V^2+gz + \frac{p}{\rho} \right) \rho V_n dA$

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Energy Conversion Equation Example

As shown in the figure below, the air passes through the 700 hp turbine. Find the flow exit velocity and the heat transferred for the conditions indicated. (R = 1716, Cp = 6003 ft.lbf/(slug.°R))

Assuming a perfect gas, the fluid density at the inlet and outlet is:

$\rho_1 = \frac{p_1}{RT_1}=\frac{150(144)}{1716(460+300)}=0.0166 \ slug/ft^3$

$\rho_2 = \frac{p_2}{RT_2}=\frac{40(144)}{1716(460+35)}=0.00679 \ slug/ft^3$

Inlet mass flow rate:

$\dot{m} = \rho_1 A_1 V_1 = (0.0166)\frac{\pi}{4}(\frac{6}{12})^2(100)=0.325 \ slug/s$

Knowing mass flow, we compute the exit velocity:

$\dot{m} = 0.325 = \rho_2 A_2 V_2 = (0.00679)\frac{\pi}{4} (\frac{6}{12})^2 V_2\Rightarrow V_2=244 \ ft/s$

Under steady state conditions, considering only the work of the shaft (Turbile), and defining the combination of internal energy and pressure terms as enthalpy (h = C_pT), the energy equation is as follows:

$\dot{Q}-\dot{W}_s = \dot{m}(C_p T_2 + \frac{1}{2} V_2^2 - C_pT_1 - \frac{1}{2} V_1^2)$

$1 \ hp = 550 \ ft.lbf/s\Rightarrow \dot{W}_s=700(550) = 385000 \ ft.lbf/s$

$\dot{Q}-385000 = 0.325 [6003(495)+\frac{1}{2}(244)^2 - 6003(760) - \frac{1}{2} (100)^2 ] = -510000 \ ft.lbf/s\Rightarrow \dot{Q} = -125000 \ ft.lbf/s$

$\dot{Q} = -125000 \ ft.lbf/s \ \frac{3600 \ s/h}{778.2 \ ft.lbf/Btu} = -578000 \ Btu/h$

Thermal Energy Conservation Equation

Many equations are used for fluid flow energy depending on the type of heat transfer. Here, one of the equations that is solved in ANSYS Fluent is reviewed.

The enthalpy of a substance (H) is the sum of sensible enthalpy (h) and latent heat (h_L):

$H=h+h_L$

Where:

$h = h_{ref} + \int_{T_{ref}}^T C_p dt$

C_p is specific heat at constant pressure and T_ref and h_ref are reference temperature and reference enthalpy, respectively. The latent heat can be written in terms of the latent heat of the material (L) and, in case of solid-liquid phase change, liquid fraction (β):

$h_L= \beta L$

For solidification and melting problems, the energy equation is expressed as:

$\frac{\partial}{\partial t}(\rho H)+\nabla.(\rho \overrightarrow{ V } H) = \nabla . (k \nabla T) +S$

Where k is the thermal conductivity coefficient, S is the source term (produced heat inside the fluid), ρ is the density and is the fluid velocity.

Mechanical Energy Conservation Equation

The equation of mechanical energy conservation, which is known as the Bernoulli equation in fluid mechanics, is:

$P+\frac{1}{2} \rho V^2 + \rho g z = constant$

Where P is the fluid pressure, ρ the density, V the fluid velocity, g the acceleration due to gravity, and z represents height above a reference level. Bernoulli’s equation can be written for two points along a streamline.

In a carburetor, by creating a flow and increasing the velocity of the fluid, the pressure in the desired area decreases, which leads to fuel being sucked into the engine. This phenomenon is described using Bernoulli’s equation.

Bernoulli Equation Assumptions

Bernoulli’s equation is based on simplifying assumptions. Although these assumptions can cause errors in Bernoulli’s results and experimental tests, the equation is still widely used in many engineering applications today. Some of these assumptions are:

Steady flow: Flow parameters and fluid properties do not change with time; note that the flow can still be turbulent.
Incompressible flow: The change in fluid density is very small.
Inviscid flow: The viscosity of the fluid is zero, or there is no friction in the system. You can reduce the error of this assumption by adding the pressure head to the equation and using the Moody diagram.
Flow along a streamline: Bernoulli’s equation describes the fluid energy along a streamline. Do not be strict about finding streamlines; instead, assume the path of the fluid. Refer to a good fluid mechanics book and solve several Bernoulli equation problems to gain confidence in assuming streamlines.
No heat transfer: Heat transfer is not considered in Bernoulli’s equation, which is a reasonable assumption for many fluid processes.
No energy addition or removal: Assume the third type of fluid energy. If a pump or turbine is in the path of the fluid, you can add or subtract the corresponding head to the equation for correction.
Conservative body forces: The only body force considered is gravity, which is a conservative force.
No phase changes: The fluid does not change phase. If the fluid changes phase, a much more complicated energy equation than Bernoulli’s is needed.

Energy Conservation Equation in ANSYS Fluent

Depending on the needs of the energy equation problem, it can have many terms, such as terms related to electromagnetic forces, phase change, turbulent energy, different heat transfer models, etc. ANSYS Fluent software handles the most complex form of the energy equation well. To solve the equation in Fluent, the equation is discretized, and in some cases, various simplification methods have been utilized to linearize it. In addition, the energy equation is solved simultaneously and coupled with other equations such as the Navier-Stokes equation, which is a complex solution process. ANSYS Fluent has demonstrated its capability to solve this energy equation effectively in numerous industrial and academic projects.

To check the energy equation and temperature changes in ANSYS Fluent, go to the Setup menu, then to Models, and enable the energy equation.

In ANSYS Fluent, in the flux reports section, you can calculate the total heat transfer rate. This option allows you to determine the amount of heat transferred from each boundary.

In CFDSHP, you can view many of our completed projects in the field of CFD using ANSYS Fluent, indicating the capabilities of our experts. You can place your CFD project orders through ORDER CFD PROJECT, where our experts will handle even the most complex simulations.

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Conclusion

Depending on the conditions of the investigated problem, this equation can be very complex or very simple. Solving this equation simultaneously with other equations and coupling them and using them in Fluid Mechanics CFD simulations algorithms is a difficult task that ANSYS Fluent does well.

Bernoulli’s equation is the most famous equation in solving fluid mechanics. Although this equation is written based on many simplifications and assumptions and is not accurate for many applications. However, even in the 21st century, this equation is used in engineering works. This equation is not directly used in CFD because it is unable to determine the details of the fluid flow.

read more:

Conservation of Mass in Fluid Mechanics

Dimensionless Numbers in Fluid Mechanics