The **drag force,** **F _{D}**, depends on the density of the fluid, the upstream velocity, and the size, shape, and orientation of the body, among other things. One way to express this is by using the

**drag equation**.

The **drag equation** is a formula used to calculate the **drag force** experienced by an object due to movement through a fluid.

The reference area, A, is defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion. For hollow objects, the reference area may be significantly larger than the cross-sectional area, but for non-hollow objects, it is the same as a cross-sectional area.

## What is Drag in Physics

In fluid dynamics, **drag** is a force acting opposite to the relative motion of any moving object. The force a flowing fluid exerts on a body in the flow direction. Unlike other resistive forces, such as dry friction, nearly independent of velocity, **drag forces** depend on velocity. **Drag force** is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. **Drag** is generally caused by two phenomena:

Skin Friction. In general, when a fluid flows over a

**stationary surface**, e.g., the flat plate, the bed of a river, or the pipe wall, the fluid touching the surface is brought**to rest**by the**shear stress**at the wall. The boundary layer is the region in which flow adjusts from zero velocity at the wall to a maximum in the mainstream of the flow. Therefore, a moving fluid exerts tangential shear forces on the surface because of the**no-slip condition**caused by viscous effects. This type of**drag force**depends especially on the geometry, the roughness of the solid surface, and the type of fluid flow.- Form Drag.
**Form drag,**also known as**pressure drag,**arises because of the shape and size of the object. This type of drag force is an interesting consequence the**Bernoulli’s effect.**According to Bernoulli’s principle, faster-moving air exerts less pressure, and this causes that there can be a pressure difference between surfaces of the object. The general size and shape of the body are the most important factors in**form drag**. Generally, bodies with a larger presented geometric cross-section will have higher drag than thinner bodies.

Both of these forces, in general, have components in the direction of flow, and thus the resulting **drag force** is due to the combined effects of **pressure** and **skin friction** forces in the flow direction.

When the friction and pressure drag coefficients are available, the total drag coefficient is determined by simply adding them:

Most drag is due to **friction drag **at **low Reynolds numbers**, especially for highly streamlined bodies such as airfoils. On the other hand, the pressure drop is significant at a** high Reynolds number**, which increases form drag.

The components of the pressure and skin friction forces in the **normal direction** to flow tend to move the body in that direction, and their sum is called** lift**.

The** lift** is an upward-acting force on an aircraft wing or **airfoil **in aeronautics**. Bernoulli’s principle** requires airfoil to be of an **asymmetrical shape**.

## Drag Force in Nuclear Engineering

Analysis of the **hydraulic lift force** is one of the most important analyses in designing a fuel assembly and analyzing the hydraulic compatibility of mixed cores. The vertical forces are induced by upward high-velocity flow through the reactor core, and theflow path for the reactor coolant through the reactor vessel would be:

The coolant enters the

**reactor vessel**at the inlet nozzle and hits against the core barrel.- The core barrel forces the water to flow downward in the space between the reactor vessel wall and the core barrel. Thisspace is usually known as the
**downcomer**. - The flow is reversed up through the core from the bottom of the pressure vessel to pass through the fuel assemblies, where the coolant temperature increases as it passes through the fuel rods.
- Finally, the hotter reactor coolant enters the upper internals region, where it is routed out the outlet nozzle into the hot legs of the primary circuit and goes on to the
**steam generators.**

Fuel assemblies are held by the **upper guide structure assembly**, which defines the top of the core. This assembly is made of stainless steel and has many purposes. The upper guide structure assembly exerts an **axial force** on fuel assemblies (through springs in the top nozzle), thus defining the exact position of the fuel assembly in the core. The **upper guide structure assembly** flange is held in place and preloaded by the RPV closure head flange. The upper guide structure assembly also guides and protects control rod assemblies and in-core instrumentation.

Required downforce of the **upper guide structure assembly **on fuel assemblies must be carefully calculated. Insufficient downforce can result in the **lift of the fuel assembly**. Onthe other hand, an excessive downforce can result in **bowing of fuel assembly**, which is also unacceptable.

## Example: Drag Force – Drag Coefficient – Fuel Bundle

Calculate the **friction drag** of a **single fuel rod** inside a reactor core at normal operation (design flow rate). Assume that this fuel rod is part of a fuel bundle with the rectangular fuel lattice, and this fuel bundle does not contain spacing grids. Its height is **h = 4m,** and the core flow velocity is constant and equal to **V**_{core}** = 5 m/s.**

Assume that:

- the outer diameter of the cladding is:
**d = 2 x r**_{Zr,1}**= 9,3 mm** - the pitch of fuel pins is:
**p = 13 mm** - the relative roughness is
**ε/D = 5×10**^{-4} - the fluid density is:
**ρ = 714 kg/m**^{3} - the core flow velocity is constant and equal to
**V**_{core}**= 5 m/s** - the average temperature of reactor coolant is:
**T**_{bulk}**= 296°C**

## Calculation of the Reynolds number

To calculate the Reynolds number, we have to know:

- the outer diameter of the cladding is:
**d = 2 x r**_{Zr,1}**= 9,3 mm**(to calculate the hydraulic diameter) - the pitch of fuel pins is:
**p = 13 mm**(to calculate the hydraulic diameter) - the dynamic viscosity of saturated water at 300°C is:
**μ = 0.0000859 N.s/m**^{2} - the fluid density is:
**ρ = 714 kg/m**^{3}

**The hydraulic diameter, D**** _{h}**, is a commonly used term when handling flow in

**non-circular tubes and channels**. The

**hydraulic diameter of the fuel channel**,

*D*

*, is equal to*

_{h}**13,85 mm**.

See also: Hydraulic Diameter

The **Reynolds number **inside the fuel channel is then equal to:

This fully satisfies the turbulent conditions.

## Calculation of the Skin Friction Coefficient

**The friction factor** for turbulent flow depends strongly on the **relative roughness. **It is determined by the Colebrook equation or can be determined using the Moody chart. The **Moody chart **for **Re = 575 600 **and **ε/D = 5 x 10**** ^{-4}** returns following values:

- the Darcy friction factor is equal to
**f**_{D}**= 0.017** - the Fanning friction factor is equal to
**f**_{F}**= f**_{D}**/4 = 0.00425**

Therefore the skin friction coefficient is equal to:

## Calculation of the Drag Force

To calculate the **drag force**, we have to know:

- the skin friction coefficient, which is:
**C**_{D,friction}**= 0.00425** - the area of pin surface, which is:
**A = π.d.h = 0.1169 m**^{2} - the fluid density, which is:
**ρ = 714 kg/m**^{3} - the core flow velocity, which is constant and equal to
**V**_{core}**= 5 m/s**

From the **skin friction coefficient, **which is equal to the Fanning friction factor, we can calculate the frictional component of the **drag force. **The drag force is given by:

Assuming that a fuel assembly can have, for example, 289 fuel pins (17×17 fuel assembly), the** frictional component** of the **drag force **is then of the order of **kilonewtons**. Moreover, this drag force originates purely from the skin friction on the fuel bundle. But typical PWR fuel assembly contains other components which influence the fuel assembly hydraulics:

**Fuel rods**. Fuel rods contain fuel and burnable poisons.**Top nozzle**. Provides the mechanical support for the fuel assembly structure.**Bottom nozzle**. Provides the mechanical support for the fuel assembly structure.**Spacing grid**. Ensures an exact guiding of the fuel rods.**Guide thimble tube**. Vacant tube for control rods or in-core instrumentation.

As was written, the second component of the drag force is the form drag. **Form drag,** also known as **pressure drag,** arises because of the **shape** and **size** of the object. The **pressure drag** is proportional to the difference between the pressures acting on the front and back of the immersed body and the frontal area.

References:

**Reactor Physics and Thermal Hydraulics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading,MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
- W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
- Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering,Springer; 4th edition, 1994, ISBN:978-0412985317
- Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
- Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
- Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition,John Wiley & Sons, 2006, ISBN:978-0-470-03037-0
- Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010,ISBN 978-1-4020-8670-0.
- U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2, and 3. June 1992.
- White Frank M., Fluid Mechanics, McGraw-Hill Education, 7th edition, February, 2010, ISBN:978-0077422417

## FAQs

### How do you calculate drag force? ›

The drag equation states that **drag D is equal to the drag coefficient Cd times the density r times half of the velocity V squared times the reference area A**.

**What is drag force equal to? ›**

The net external force is equal to the difference between the weight and the drag forces (**F = W - D**). The acceleration of the object then becomes a = (W - D) / m . The drag force depends on the square of the velocity. So as the body accelerates its velocity (and the drag) will increase.

**What is drag force in physics? ›**

In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is **a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid**.

**Why is there a 1/2 in the drag equation? ›**

By the work/kinetic-energy theorem, the work done is equal to the change in kinetic energy that the object experiences. Since **kinetic energy is defined as $E_k = 1/2 mv^2$**, you can expect the "1/2" and the $v^2$ terms to show up in the equation.

**What is K in the drag equation? ›**

where C_{D0} represents the friction and pressure drag components and K ( C L − C L 0 ) 2 , the **lift dependent components**.

**What is drag force with example? ›**

In commonly used context drag force is **the force that is exerted on a solid body moving with respect to a fluid due to the movement of the fluid**. For example drag on a ship moving in water or drag on a plane moving in the air.

**What are the 2 drag forces? ›**

Drag force can be broken into two types: **form drag and skin drag** . Form drag is caused by the resistance of fluids (liquids or gases) to being pushed out of the way by an object in motion through the fluid.

**How do you calculate B in drag force? ›**

Drag Forces. where C is the drag coefficient, A is the area of the object facing the fluid, and ρ is the density of the fluid. (Recall that density is mass per unit volume.) This equation can also be written in a more generalized fashion as **FD=bv2**, where b is a constant equivalent to 0.5CρA.

**What are the units of drag force? ›**

Variable | Identity | Metric Units |
---|---|---|

D | Drag | Newtons |

Cd | Drag Coefficient | No units |

r | Density of air | kg/m^{3} |

V | Velocity | m/s |

**Who created the drag equation? ›**

**Rayleigh** "derived" the drag equation in On the Resistance of Fluids, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser. 5, v. 2 (1876) no. 13, 430-441.

### Is drag force constant? ›

(Note that, due to the way the filters are nested, **drag is constant** and only mass varies.) They obtain terminal velocity quite quickly, so find this velocity as a function of mass. Plot the terminal velocity v versus mass.

**What is B in drag? ›**

(3) Theory states that this constant “b” is composed of the density of the fluid through which the object is. moving (in this case, air), the cross-sectional area of the object (in this case, the coffee filters), and a. “**drag coefficient**” (a unitless constant which depends on the shape of the object) which yields a ...

**What is the drag coefficient value? ›**

The drag coefficient (non-dimensional drag) is **equal to the drag force divided by the product of velocity pressure and frontal area**. The velocity may be that of the object through the air (or any other gas) or the air velocity past a stationary object.

**How do you calculate drag coefficient from Reynolds number? ›**

Figure 1 graphs the dependence of drag coefficient for a sphere and a cylinder in crossflow on the Reynolds Number **Re = ρuD/η**, where D is the sphere (cylinder) diameter, η the viscosity of liquid, and .

**Does drag force depend on mass? ›**

**Drag depends directly on the mass of the flow going past the aircraft**. The drag also depends in a complex way on two other properties of the air: its viscosity and its compressibility. These factors affect the wave drag and skin friction which are described above.

**What is K in induced drag? ›**

Induced drag factor K = **(1/πeA)** • Planform factor e = 4.61 ((1 – 0.045A^{0.68}) (cos Λ)^{0.15}) – 3.1. • Assumed skin friction coefficient C_{f} = 0.0038.

**What is K in aerodynamics? ›**

For aircraft fuel flow meters, K-factor refers to **the number of pulses expected for every one volumetric unit of fluid passing through a given flow meter**, and is usually encountered when dealing with pulse signals.

**What is drag function? ›**

Drag functions are **a measure of the drag of a "standard" bullet**. This standard bullet has a C of 1.0. Bullets of the same shape typically have a drag curves (drag as a function of speed) that are the same or very similar.

**How do you calculate drag force on a moving object? ›**

**D = (C**

_{d}× ρ × V^{2}× A) / 2- D is the Drag Force (N),
- C
_{d}is the Drag coefficient, - ρ is the Density of the medium (kg/m
^{3}), - V is the Velocity of the body (m/s), and.
- A is the Cross-sectional area (m
^{2}).

**What is drag force simple? ›**

Drag is **the aerodynamic force that opposes an aircraft's motion through the air**.

### How do you calculate drag on a plane? ›

The drag equation states that **drag (D)is equal to a drag coefficient (Cd) times the density of the air (r) times half of the square of the velocity (V) times the wing area (A)**.

**How is drag force measured experimentally? ›**

Drag force measurements on various bodies can be obtained **using a subsonic wind tunnel**, which can be found in most laboratories. Making measurements of drag force versus velocity using spheres, hemispheres, disks, and flat plates are classical experiments.

**How do you calculate drag force on a sphere? ›**

He found what has become known as Stokes' Law: the drag force F on a sphere of radius a moving through a fluid of viscosity η at speed v is given by: **F=6πaηv**. Note that this drag force is directly proportional to the radius.

**What determines drag? ›**

The drag equation. is essentially a statement that the drag force on any object is proportional to the density of the fluid and proportional to the square of the relative flow speed between the object and the fluid. **The factor of**. **comes from the dynamic pressure of the fluid, which is equal to the kinetic energy** ...

**How does drag force work? ›**

**Drag is generated by the difference in velocity between the solid object and the fluid**. There must be motion between the object and the fluid. If there is no motion, there is no drag. It makes no difference whether the object moves through a static fluid or whether the fluid moves past a static solid object.

**What does drag force depend on? ›**

Drag depends directly on **the mass of the flow going past the aircraft**. The drag also depends in a complex way on two other properties of the air: its viscosity and its compressibility. These factors affect the wave drag and skin friction which are described above.

**What is B in drag? ›**

(3) Theory states that this constant “b” is composed of the density of the fluid through which the object is. moving (in this case, air), the cross-sectional area of the object (in this case, the coffee filters), and a. “**drag coefficient**” (a unitless constant which depends on the shape of the object) which yields a ...

**How do you calculate drag coefficient using Reynolds number? ›**

Figure 1 graphs the dependence of drag coefficient for a sphere and a cylinder in crossflow on the Reynolds Number **Re = ρuD/η**, where D is the sphere (cylinder) diameter, η the viscosity of liquid, and .

**What are the units of drag force? ›**

Variable | Identity | Metric Units |
---|---|---|

D | Drag | Newtons |

Cd | Drag Coefficient | No units |

r | Density of air | kg/m^{3} |

V | Velocity | m/s |

**What is the drag of a sphere? ›**

Drag coefficient of sphere is very important factor for a matter. Drag coefficient of sphere derive as, **the ratio between the surface area of the sphere of the similar volume for the matter of the body comparative to the surface area for the matter of the body**.

### What is drag coefficient of sphere? ›

Type of Object | Drag Coefficient - c_{d} - | Frontal Area |
---|---|---|

Solid Hemisphere | 0.42 | π / 4 d2 |

Sphere | 0.5 | |

Saloon Car, stepped rear | 0.4 - 0.5 | frontal area |

Bike - Drafting behind an other cyclist | 0.5 | 3.9 ft^{2} (0.36 m^{2}) |

**What are the 2 drag forces? ›**

Drag force can be broken into two types: **form drag and skin drag** . Form drag is caused by the resistance of fluids (liquids or gases) to being pushed out of the way by an object in motion through the fluid.

**How do you calculate drag force on a moving object? ›**

**D = (C**

_{d}× ρ × V^{2}× A) / 2- D is the Drag Force (N),
- C
_{d}is the Drag coefficient, - ρ is the Density of the medium (kg/m
^{3}), - V is the Velocity of the body (m/s), and.
- A is the Cross-sectional area (m
^{2}).

**What is drag flow? ›**

Drag flow is simply **half the volume of one turn of the metering section per second at a specific screw rpm**, which, when multiplied by a units conversion and the melt specific gravity of the polymer, is a very accurate approximation of the output in lb/hr at no head pressure.

**Who created the drag equation? ›**

**Rayleigh** "derived" the drag equation in On the Resistance of Fluids, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser. 5, v. 2 (1876) no. 13, 430-441.