Thermal Engineering

Bernoulli’s Equation

The Bernoulli’s equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. It is one of the most important/useful equations in fluid mechanics. It puts into a relation pressure and velocity in an inviscid incompressible flow. Bernoulli’s equation has some restrictions in its applicability, they summarized in following points:

• steady flow system,
• density is constant (which also means the fluid is incompressible),
• no work is done on or by the fluid,
• no heat is transferred to or from the fluid,
• no change occurs in the internal energy,
• the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, the general energy equation is simplified to:

This equation is the most famous equation in fluid dynamics. The Bernoulli’s equation describes the qualitative behavior flowing fluid that is usually labeled with the term Bernoulli’s effect. This effect causes the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy. The dimensions of terms in the equation are kinetic energy per unit volume.

Bernoulli’s Effect – Spinning ball in an airflow

The Bernoulli’s effect has another interesting interesting consequence. Suppose a ball is spinning as it travels through the air. As the ball spins, the surface friction of the ball with the surrounding air drags a thin layer (referred to as the boundary layer) of air with it. It can be seen from the picture the boundary layer is on one side traveling in the same direction as the air stream that is flowing around the ball (the upper arrow) and on the other side, the boundary layer is traveling in the opposite direction (the bottom arrow). On the side of the ball where the air stream and boundary layer are moving in the opposite direction (the bottom arrow) to each other friction between the two slows the air stream. On the opposite side these layers are moving in the same direction and the stream moves faster.

According to Bernoulli’s principle, faster moving air exerts less pressure, and therefore the air must exert an upward force on the ball. In fact, in this case the use of Bernoulli’s principle may not be correct. The Bernoulli’s principle assumes incompressibility of the air, but in reality the air is easily compressible. But there are more limitations of explanations based on Bernoulli’s principle.

The work of Robert G. Watts and Ricardo Ferrer (The lateral forces on a spinning sphere: Aerodynamics of a curveball) this effect can be explained by another model which gives important attention to the spinning boundary layer of air around the ball. On the side of the ball where the air stream and boundary layer are moving in the opposite direction (the bottom arrow), the boundary layer tends to separate prematurely. On the side of the ball where the air stream and boundary layer are moving in the same direction , the boundary layer carries further around the ball before it separates into turbulent flow. This gives a flow deflection of the airstream in one direction behind the ball. The rotating ball generates lift by exerting a downward force on the air as it flows past. According to Newton’s third law, the air must exert an upward force on the ball.

 

See also:

Bernoulli’s Principle

Bernoulli’s Equation

The Bernoulli’s equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. It is one of the most important/useful equations in fluid mechanics. It puts into a relation pressure and velocity in an inviscid incompressible flow. Bernoulli’s equation has some restrictions in its applicability, they summarized in following points:

• steady flow system,
• density is constant (which also means the fluid is incompressible),
• no work is done on or by the fluid,
• no heat is transferred to or from the fluid,
• no change occurs in the internal energy,
• the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, the general energy equation is simplified to:

This equation is the most famous equation in fluid dynamics. The Bernoulli’s equation describes the qualitative behavior flowing fluid that is usually labeled with the term Bernoulli’s effect. This effect causes the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy. The dimensions of terms in the equation are kinetic energy per unit volume.

Torricelli’s law

Torricelli’s law, also known as Torricelli’s principle, or Torricelli’s theorem, statement in fluid dynamics that the speed, v, of fluid flowing out of an orifice under the force of gravity in a tank is proportional to the square root of the vertical distance, h, between the liquid surface and the centre of the orifice and to the square root of twice the acceleration caused by gravity (g = 9.81 N/kg near the surface of the earth).

In other words, the efflux velocity of the fluid from the orifice is the same as that it would have acquired by falling a height h under gravity. The law was discovered by and named after the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli’s principle.

The Torricelli’s equation is derived for a specific condition. The orifice must be small and viscosity and other losses must be ignored. If a fluid is flowing through a very small orifice (for example at the bottom of a large tank) then the velocity of the fluid at the large end can be neglected in Bernoulli’s Equation. Moreover the speed of efflux is independent of the direction of flow. In that case the efflux speed of fluid flowing through the orifice given by following formula:

v = √2gh

 

See also:

Bernoulli’s Principle

Conservation of Energy

This principle is generally known as the conservation of energy principle and states that the total energy of an isolated system remains constant — it is said to be conserved over time. This is equivalent to the First Law of Thermodynamics, which is used to develop the general energy equation in thermodynamics. This principle can be use in the analysis of flowing fluids and this principle is expressed mathematically by following equation:

where h is enthalpy, k is the thermal conductivity of the fluid, T is temperature, and Φ is the viscous dissipation function.

Bernoulli’s Theorem

The Bernoulli’s theorem can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. It is one of the most important/useful equations in fluid mechanics. It puts into a relation pressure and velocity in an inviscid incompressible flow. Bernoulli’s equation has some restrictions in its applicability, they summarized in following points:

• steady flow system,
• density is constant (which also means the fluid is incompressible),
• no work is done on or by the fluid,
• no heat is transferred to or from the fluid,
• no change occurs in the internal energy,
• the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, the general energy equation is simplified to:

This equation is the most famous equation in fluid dynamics. The Bernoulli’s equation describes the qualitative behavior flowing fluid that is usually labeled with the term Bernoulli’s effect. This effect causes the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy. The dimensions of terms in the equation are kinetic energy per unit volume.

 

See also:

Bernoulli’s Principle

Bernoulli’s Equation

The Bernoulli’s equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. It is one of the most important/useful equations in fluid mechanics. It puts into a relation pressure and velocity in an inviscid incompressible flow. Bernoulli’s equation has some restrictions in its applicability, they summarized in following points:

• steady flow system,
• density is constant (which also means the fluid is incompressible),
• no work is done on or by the fluid,
• no heat is transferred to or from the fluid,
• no change occurs in the internal energy,
• the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, the general energy equation is simplified to:

This equation is the most famous equation in fluid dynamics. The Bernoulli’s equation describes the qualitative behavior flowing fluid that is usually labeled with the term Bernoulli’s effect. This effect causes the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy. The dimensions of terms in the equation are kinetic energy per unit volume.

Bernoulli’s Effect – Relation between Pressure and Velocity

It is an illustrative example, following data do not correspond to any reactor design.

When the Bernoulli’s equation is combined with the continuity equation the two can be used to find velocities and pressures at points in the flow connected by a streamline.

The continuity equation is simply a mathematical expression of the principle of conservation of mass. For a control volume that has a single inlet and a single outlet, the principle of conservation of mass states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out.

Example:

Determine pressure and velocity within a cold leg of primary piping and determine pressure and velocity at a bottom of a reactor core, which is about 5 meters below the cold leg of primary piping.

Let assume:

• Fluid of constant density ⍴ ~ 720 kg/m³ (at 290°C) is flowing steadily through the cold leg and through the core bottom.

• Primary piping flow cross-section (single loop) is equal to 0.385 m² (piping diameter ~ 700mm)

• Flow velocity in the cold leg is equal to 17 m/s.

• Reactor core flow cross-section is equal to 5m².

• The gauge pressure inside the cold leg is equal to 16 MPa.

As a result of the Continuity principle the velocity at the bottom of the core is:

v_inlet =  v_cold . A_piping / A_core = 17 x 1.52 / 5 = 5.17 m/s

As a result of the Bernoulli’s principle the pressure at the bottom of the core (core inlet) is:

 

See also:

Bernoulli’s Principle

Extended Bernoulli’s Equation

There are two main assumptions, that were applied on the derivation of the simplified Bernoulli’s equation.

• The first restriction on Bernoulli’s equation is that no work is allowed to be done on or by the fluid. This is a significant limitation, because most hydraulic systems (especially in nuclear engineering) include pumps. This restriction prevents two points in a fluid stream from being analyzed if a pump exists between the two points.

• The second restriction on simplified Bernoulli’s equation is that no fluid friction is allowed in solving hydraulic problems. In reality, friction plays crucial role. The total head possessed by the fluid cannot be transferred completely and lossless from one point to another. In reality, one purpose of pumps incorporated in a hydraulic system is to overcome the losses in pressure due to friction.

Due to these restrictions most of practical applications of the simplified Bernoulli’s equation to real hydraulic systems are very limited. In order to deal with both head losses and pump work, the simplified Bernoulli’s equation must be modified.

The Bernoulli equation can be modified to take into account gains and losses of head. The resulting equation, referred to as the extended Bernoulli’s equation, is very useful in solving most fluid flow problems. The following equation is one form of the extended Bernoulli’s equation.

where:
h = height above reference level (m)
v = average velocity of fluid (m/s)
p = pressure of fluid (Pa)
H_pump = head added by pump (m)
H_friction = head loss due to fluid friction (m)
g = acceleration due to gravity (m/s²)

The head loss (or the pressure loss) due to fluid friction (H_friction) represents the energy used in overcoming friction caused by the walls of the pipe. The head loss that occurs in pipes is dependent on the flow velocity, pipe diameter and length, and a friction factor based on the roughness of the pipe and the Reynolds number of the flow. A piping system containing many pipe fittings and joints, tube convergence, divergence, turns, surface roughness and other physical properties will also increase the head loss of a hydraulic system.

Although the head loss represents a loss of energy, it does does not represent a loss of total energy of the fluid. The total energy of the fluid conserves as a consequence of the law of conservation of energy. In reality, the head loss due to friction results in an equivalent increase in the internal energy (increase in temperature) of the fluid.

Most methods for evaluating head loss due to friction are based almost exclusively on experimental evidence. This will be discussed in following sections.

 

 

See also:

Bernoulli’s Principle

Hydraulic Head

In general, the hydraulic head, or total head, is a measure of the potential of fluid at the measurement point. It can be used to determine a hydraulic gradient between two or more points.

In fluid dynamics, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. The units for all the different forms of energy in the Bernoulli’s equation can be measured also in units of distance, and therefore these terms are sometimes referred to as “heads” (pressure head, velocity head, and elevation head). Head is also defined for pumps. This head is usually referred to as the static head and represents the maximum height (pressure) it can deliver. Therefore the characteristics of all pumps can be usually read from its Q-H curve (flow rate – height).

There are four types of potential (head):

• Pressure potential – Pressure head: The pressure head represents the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid.ρ_w: density of water assumed to be independent of pressure
• Elevation potential – Elevation head: The elevation head represents the potential energy of a fluid due to its elevation above a reference level.
• Kinetic potential – Kinetic head: The kinetic head represents the kinetic energy of the fluid. It is the height in feet that a flowing fluid would rise in a column if all of its kinetic energy were converted to potential energy.

The sum of the elevation head, kinetic head, and pressure head of a fluid is called the total head. Thus, Bernoulli’s equation states that the total head of the fluid is constant.

  Consider a pipe containing an ideal fluid. If this pipe undergoes a gradual expansion in diameter, the continuity equation tells us that as the pipe diameter increases, the flow velocity must decrease in order to maintain the same mass flow rate. Since the outlet velocity is less than the inlet velocity, the kinetic head of the flow must decrease from the inlet to the outlet. If there is no change in elevation head (the pipe lies horizontal), the decrease in kinetic head must be compensated for by an increase in pressure head.

 

 

See also:

Bernoulli’s Principle

Pressure Head

In general, the hydraulic head, or total head, is a measure of the potential of fluid at the measurement point. It can be used to determine a hydraulic gradient between two or more points.

In fluid dynamics, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. The units for all the different forms of energy in the Bernoulli’s equation can be measured also in units of distance, and therefore these terms are sometimes referred to as “heads” (pressure head, velocity head, and elevation head). Head is also defined for pumps. This head is usually referred to as the static head and represents the maximum height (pressure) it can deliver. Therefore the characteristics of all pumps can be usually read from its Q-H curve (flow rate – height).

There are four types of potential (head):

• Pressure potential – Pressure head: The pressure head represents the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid.ρ_w: density of water assumed to be independent of pressure
• Elevation potential – Elevation head: The elevation head represents the potential energy of a fluid due to its elevation above a reference level.
• Kinetic potential – Kinetic head: The kinetic head represents the kinetic energy of the fluid. It is the height in feet that a flowing fluid would rise in a column if all of its kinetic energy were converted to potential energy.

The sum of the elevation head, kinetic head, and pressure head of a fluid is called the total head. Thus, Bernoulli’s equation states that the total head of the fluid is constant.

  Consider a pipe containing an ideal fluid. If this pipe undergoes a gradual expansion in diameter, the continuity equation tells us that as the pipe diameter increases, the flow velocity must decrease in order to maintain the same mass flow rate. Since the outlet velocity is less than the inlet velocity, the kinetic head of the flow must decrease from the inlet to the outlet. If there is no change in elevation head (the pipe lies horizontal), the decrease in kinetic head must be compensated for by an increase in pressure head.

 

 

See also:

Bernoulli’s Principle

Velocity Head

In general, the hydraulic head, or total head, is a measure of the potential of fluid at the measurement point. It can be used to determine a hydraulic gradient between two or more points.

In fluid dynamics, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. The units for all the different forms of energy in the Bernoulli’s equation can be measured also in units of distance, and therefore these terms are sometimes referred to as “heads” (pressure head, velocity head, and elevation head). Head is also defined for pumps. This head is usually referred to as the static head and represents the maximum height (pressure) it can deliver. Therefore the characteristics of all pumps can be usually read from its Q-H curve (flow rate – height).

There are four types of potential (head):

• Pressure potential – Pressure head: The pressure head represents the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid.ρ_w: density of water assumed to be independent of pressure
• Elevation potential – Elevation head: The elevation head represents the potential energy of a fluid due to its elevation above a reference level.
• Kinetic potential – Kinetic head: The kinetic head represents the kinetic energy of the fluid. It is the height in feet that a flowing fluid would rise in a column if all of its kinetic energy were converted to potential energy.

The sum of the elevation head, kinetic head, and pressure head of a fluid is called the total head. Thus, Bernoulli’s equation states that the total head of the fluid is constant.

  Consider a pipe containing an ideal fluid. If this pipe undergoes a gradual expansion in diameter, the continuity equation tells us that as the pipe diameter increases, the flow velocity must decrease in order to maintain the same mass flow rate. Since the outlet velocity is less than the inlet velocity, the kinetic head of the flow must decrease from the inlet to the outlet. If there is no change in elevation head (the pipe lies horizontal), the decrease in kinetic head must be compensated for by an increase in pressure head.

 

 

See also:

Bernoulli’s Principle

Elevation Head

In general, the hydraulic head, or total head, is a measure of the potential of fluid at the measurement point. It can be used to determine a hydraulic gradient between two or more points.

In fluid dynamics, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. The units for all the different forms of energy in the Bernoulli’s equation can be measured also in units of distance, and therefore these terms are sometimes referred to as “heads” (pressure head, velocity head, and elevation head). Head is also defined for pumps. This head is usually referred to as the static head and represents the maximum height (pressure) it can deliver. Therefore the characteristics of all pumps can be usually read from its Q-H curve (flow rate – height).

There are four types of potential (head):

• Pressure potential – Pressure head: The pressure head represents the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid.ρ_w: density of water assumed to be independent of pressure
• Elevation potential – Elevation head: The elevation head represents the potential energy of a fluid due to its elevation above a reference level.
• Kinetic potential – Kinetic head: The kinetic head represents the kinetic energy of the fluid. It is the height in feet that a flowing fluid would rise in a column if all of its kinetic energy were converted to potential energy.

The sum of the elevation head, kinetic head, and pressure head of a fluid is called the total head. Thus, Bernoulli’s equation states that the total head of the fluid is constant.

  Consider a pipe containing an ideal fluid. If this pipe undergoes a gradual expansion in diameter, the continuity equation tells us that as the pipe diameter increases, the flow velocity must decrease in order to maintain the same mass flow rate. Since the outlet velocity is less than the inlet velocity, the kinetic head of the flow must decrease from the inlet to the outlet. If there is no change in elevation head (the pipe lies horizontal), the decrease in kinetic head must be compensated for by an increase in pressure head.

 

 

See also:

Bernoulli’s Principle

Head Loss – Pressure Loss

In the practical analysis of piping systems the quantity of most importance is the pressure loss due to viscous effects along the length of the system, as well as additional pressure losses arising from other technological equipments like, valves, elbows, piping entrances, fittings and tees.
At first, an extended Bernoulli’s equation must be introduced. This equation permits account of viscosity to be included in an empirical way and quantify this with a physical parameter known as the head loss.

The head loss (or the pressure loss) represents the reduction in the total head or pressure (sum of elevation head, velocity head and pressure head) of the fluid as it flows through a hydraulic system. The head loss also represents the energy used in overcoming friction caused by the walls of the pipe and other technological equipments. The head loss is unavoidable in real moving fluids. It is present because of the friction between adjacent fluid particles as they move relative to one another (especially in turbulent flow).

The head loss that occurs in pipes is dependent on the flow velocity, pipe diameter and length, and a friction factor based on the roughness of the pipe and the Reynolds number of the flow. Although the head loss represents a loss of energy, it does not represent a loss of total energy of the fluid. The total energy of the fluid conserves as a consequence of the law of conservation of energy. In reality, the head loss due to friction results in an equivalent increase in the internal energy (increase in temperature) of the fluid.

Most methods for evaluating head loss due to friction are based almost exclusively on experimental evidence. This will be discussed in following sections.

Classification of Head Loss

The head loss of a pipe, tube or duct system, is the same as that produced in a straight pipe or duct whose length is equal to the pipes of the original systems plus the sum of the equivalent lengths of all the components in the system.

As can be seen, the head loss of piping system is divided into two main categories, “major losses” associated with energy loss per length of pipe, and “minor losses” associated with bends, fittings, valves, etc.

• Major Head Loss – due to friction in pipes and ducts.
• Minor Head Loss – due to components as valves, fittings, bends and tees.

The head loss can be then expressed as:

h_loss = Σ h_{major_losses} + Σ h_{minor_losses}

Summary:

• Head loss or pressure loss are the reduction in the total head (sum of potential head, velocity head, and pressure head) of a fluid caused by the friction present in the fluid’s motion.
• Head loss and pressure loss represent the same phenomenon – frictional losses in pipe and losses in hydraulic components, but they are expressed in different units.
• Head loss of hydraulic system is divided into two main categories:
  • Major Head Loss – due to friction in straight pipes
  • Minor Head Loss – due to components as valves, bends…
• Darcy’s equation can be used to calculate major losses.
• A special form of Darcy’s equation can be used to calculate minor losses.
• The friction factor for fluid flow can be determined using a Moody chart.

Why the head loss is very important?

As can be seen from the picture, the head loss is forms key characteristic of any hydraulic system. In systems, in which some certain flowrate must be maintained (e.g. to provide sufficient cooling or heat transfer from a reactor core), the equilibrium of the head loss and the head added by a pump determines the flowrate through the system.

Major Head Loss – Frictional Loss

See also: Major Head Loss – Frictional Losses

Major losses, which are associated with frictional energy loss per length of pipe depends on the flow velocity, pipe length, pipe diameter, and a friction factor based on the roughness of the pipe, and whether the flow is laminar or turbulent (i.e. the Reynolds number of the flow).

Although the head loss represents a loss of energy, it does not represent a loss of total energy of the fluid. The total energy of the fluid conserves as a consequence of the law of conservation of energy. In reality, the head loss due to friction results in an equivalent increase in the internal energy (increase in temperature) of the fluid.

By observation, the major head loss is roughly proportional to the square of the flow rate in most engineering flows (fully developed, turbulent pipe flow).

The most common equation used to calculate major head losses in a tube or duct is the Darcy–Weisbach equation (head loss form).

where:

• Δh = the head loss due to friction (m)
• f_D = the Darcy friction factor (unitless)
• L = the pipe length (m)
• D = the hydraulic diameter of the pipe D (m)
• g = the gravitational constant (m/s²)
• V = the mean flow velocity V (m/s)

  Evaluating the Darcy-Weisbach equation provides insight into factors affecting the head loss in a pipeline.

• Consider that the length of the pipe or channel is doubled, the resulting frictional head loss will double.
• At constant flow rate and pipe length, the head loss is inversely proportional to the 4th power of diameter (for laminar flow), and thus reducing the pipe diameter by half increases the head loss by a factor of 16. This is a very significant increase in head loss, and shows why larger diameter pipes lead to much smaller pumping power requirements.
• Since the head loss is roughly proportional to the square of the flow rate, then if the flow rate is doubled, the head loss increases by a factor of four.
• The head loss is reduced by half (for laminar flow) when the viscosity of the fluid is reduced by half.

With the exception of the Darcy friction factor, each of these terms (the flow velocity, the hydraulic diameter, the length of a pipe) can be easily measured. The Darcy friction factor takes the fluid properties of density and viscosity into account, along with the pipe roughness. This factor may be evaluated by the use of various empirical relations, or it may be read from published charts (e.g. Moody chart).

 

Minor Head Loss – Local Pressure Loss

See also: Minor Head Loss – Local Pressure Loss

In industry any pipe system contains different technological elements as bends, fittings, valves or heated channels. These additional components add to the overall head loss of the system. Such losses are generally termed minor losses, although they often account for a major portion of the head loss. For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses (especially with a partially closed valve that can cause a greater pressure loss than a long pipe, in fact when a valve is closed or nearly closed, the minor loss is infinite).

The minor losses are commonly measured experimentally. The data, especially for valves, are somewhat dependent upon the particular manufacturer’s design.

Generally, most of methods that are used in industry, define a coefficient K as a value for certain technological component.

Like pipe friction, the minor losses are roughly proportional to the square of the flow rate and therefore they can be easy integrated into the Darcy-Weisbach equation. K is the sum of all of the loss coefficients in the length of pipe, each contributing to the overall head loss.

The following methods are of practical importance in local pressure loss calculations:

• Equivalent Length Method
• K-Method – Resistance Coefficient Method
• 2K-Method
• 3K-Method

See also: Minor Head Loss – Local Pressure Loss

 

 

Head Loss of Two-phase Fluid Flow

See also: Two-phase Pressure Drop

In contrast to single-phase pressure drops, calculation and prediction of two-phase pressure drops is much more sophisticated problem and leading methods differ significantly. Experimental data indicates that the frictional pressure drop in the two-phase flow (e.g. in a boiling channel) is substantially higher than that for a single-phase flow with the same length and mass flow rate. Explanations for this include an apparent increased surface roughness due to bubble formation on the heated surface and increased flow velocities.

 

See also:

Bernoulli’s Principle