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## Hydraulic diameter definition

Circle being the simplest shape, easiest form of calculations come around while dealing with circular cross sections. When fluid flows through a non-circular duct, we convert the cross section to circular for convenient calculations. This newly derived diameter of circular cross section is called as hydraulic diameter. It is denoted as D_{h}. Hence, we can find the same results for a non-circular duct as circular duct by using the concept of hydraulic diameter.

## Hydraulic diameter equation

Hydraulic diameter can be found using the formula given below-

Dh = 4A/P

Where,

D_{h }is hydraulic diameter

A is area of non-circular cross section

P is the wetted perimeter of non-circular cross section

Hydraulic diameter is a function of hydraulic radius R_{h}, which can be found by dividing area of cross section, A by wetted perimeter, P.

Note that D_{h }= 4R_{h}This relation is different from the conventional relation between diameter and radius (i.e. D = 2R). This difference arises only while converting non-circular cross sections to circular.

Note- Law of conservation of momentum is satisfied while calculating the hydraulic diameter. Also, hydraulic diameter is not same as normal diameter. D_{h} is same only for circular conduits.

## Hydraulic diameter and Reynold’s number

Reynold’s number is used in fluid mechanics and heat transfer to find the type of flow, laminar or turbulent. Hydraulic diameter is used in the formula to calculate Reynold’s number.

Reynold’s number is the ratio inertia forces to viscous forces. It is a dimensionless number named after Irish scientist Osborne Reynolds who popularized this concept in 1883.

This number shows the effect of viscosity in controlling the velocity of flowing fluid. A linear profile of viscosity is developed when the flow is laminar. In Laminar flow, the fluid flows in such a way that it appears as if it was flowing in parallel layers. These layers do not intersect each other and move without any disruption in between them. This type of flow usually occurs at slow speeds. At slow speeds, mixing of two layers doesn’t take place and fluid flows in layers stacked above one another.

Laminar flow helps us to measure the flow of highly viscous fluids as this type of flow gives a linear relationship between flow rate and pressure drop. Favorable conditions for laminar flow is high viscosity and low velocity. At greater speeds, the fluid particles start behaving in a different manner resulting in mixing of fluid layers. Such mixing gives rise to turbulence and hence the name turbulent flow. Turbulent flow is desirable when proper mixing of fluid is required. One such example is mixing of fuel with oxidizer in rocket engines. Turbulence helps in thorough mixing of fluid.

Reynold’s number can be calculated from the equation given below-

Where,

Re is Reynold’s number

u is mean speed velocity (in m/s)

ν is kinematic viscosity (in m^{2}/s)

Dh is hydraulic diameter (in m)

In a circular pipe,

Laminar flow, Re < 2000

Transient flow, 2000 < Re <4000

Turbulent flow, Re > 4000

For a flat plate,

Laminar flow, Re <5,00,000

Turbulent flow, Re > 5,00,000

## Hydraulic diameter of circular pipe | hydraulic diameter of cylinder

Circular pipes are most commonly used pipes for transporting fluid/gas from one place to other (even for large distances). Water pipelines are real life example of circular ducts that are used for transporting fluid. These pipes can carry large distances such as from water filter stations to homes as well as short distances such as ground water tank to terrace water tank. The hydraulic diameter of circular pipe is given by-

Dh = 4πR^{2}/2πR = 2R

Where,

R is the radius of circular cross section.

**Hydraulic diameter of rectangular duct**

Rectangular ducts are used when spacing is an issue. Moreover, rectangular ducts are easy to fabricate and reduce pressure loss. Air conditioners use rectangular ducts to avoid pressure losses. The hydraulic diameter of rectangular duct is given by-

Dh = 4ab/2(a+b) = 2ab/ a+b

Where,

a and b are the lengths of larger and shorter sides.

For square cross section,

a = b

Dh = 2a^{2}/2a = a

Where,

a is the length of each side of square.

## Hydraulic diameter of annulus

Sometimes, to increase/decrease the rate of heat transfer, two fluids are passed through an annular tube such that one fluid flows outside the other. heat transfer rate is affected by the action of two fluids. Hydraulic diameter of annulus is given by-

Where D and d are diameters of outer circle and inner circle respectively.

## Hydraulic diameter of triangle

Where,

l is the length of each side.

## Hydraulic diameter of ellipse

Dh = 4*wh*(64-16e^{2})/*w+h*(64-3e^{4})

Where, e=* w-h*/*w+h*

## Hydraulic diameter of plate heat exchanger | hydraulic diameter of shell and tube heat exchanger

Heat exchangers are thermal devices used for transferring heat from one fluid to other in order to decrease/increase the temperature of fluid as desired. Many types of heat exchangers exist out of which most commonly used are plate and shell tube heat exchangers. Fluids can be passed through the heat exchanger in two ways. In first type, both hot and cold fluids are injected in the same direction hence, it is called as parallel flow heat exchanger. In second type, fluids are passed through the tube in opposite directions hence it is called as a counter flow heat exchanger.

Based on this, evaporator and condenser are designed. In evaporator, the hot fluid’s temperature remains same while the cold fluid gets warmer. In condenser, the temperature of cold fluid remains same and hotter fluid’s temperature decreases.

The rate of transfer in heat exchanger is given by following relation-

For hot fluid: Q_{h} = m_{h} C_{ph} (T_{hi} – T_{ho} )

For cold fluid: Q_{c} = m_{c }C_{pc} (T_{co} – T_{ci} )

By conservation of energy,

Heat lost by hot fluid = heat gained by cold fluid.

=> Q_{h} = Q_{c}Where,

Q_{h} denotes heat lost by hot fluid

Q_{c} denotes the heat gained by cold fluid

T_{hi} is the temperature of hot fluid at inlet

T_{ho} is the temperature of hot fluid at outlet

T_{ci} is the temperature of cold fluid at inlet

T_{co} is the temperature of cold fluid at outlet

m_{h} is the mass of hot fluid (in Kg)

m_{c} is the mass of cold fluid (in Kg)

C_{ph} is the specific heat of hot fluid (in J/K-Kg)

C_{pc} is the specific heat of cold fluid (in J/K-Kg)

In plate heat exchangers, heat cuts through the section and separates hot and cold fluids. This type of heat exchanger is used in many industrial applications. They are used in heat pump, oil cooling systems, engine cooling system, thermal storage systems etc.

Plate heat exchanger has a rectangular/square cross section hence, hydraulic diameter is given by-

Dh = 2ab/a+b

Where,

a and b are lengths of shorter side and longer side respectively.

In shell and tube type heat exchanger, tubes are installed in a cylindrical shell. Both hot and cold fluids are passed through these tubes in such a way that one fluid flows outside the other fluid. Due to this, heat is transferred from one fluid to another. Shell type heat exchanger is widely used in industries mainly in chemical processes and applications where high pressure is needed.

Shell tube heat exchanger has annular cross section hence, hydraulic diameter is given by

D_{h }= D-d

Shell and tube heat exchanger

Image credits: Straight-tube heat exchanger 2-pass

## Equivalent diameter vs Hydraulic diameter

Equivalent diameter and hydraulic diameter differ in values. The diameter of circular duct which gives same pressure loss as rectangular duct for equal flow is called as equivalent diameter. Even though circular ducts have least surface area for given pressure loss, they are not suitable for fabrication. Rectangular ducts are easy to fabricate hence they are used in practical cases. When flow rate and pressure drop is known, then to design a rectangular duct, we use friction chart to find the equivalent diameter and then required dimensions by fixing certain parameters like aspect ratio or length of any one side.

The ratio of length of shorter side to longer side is called as aspect ratio.

AR = a/b

We can find equivalent diameter by Huebscher equivalent diameter equation. It is shown below-

De = 1.30 (ab)^{0.625}/(a+b)^{0.25}

Where,

a and b are length of shorter side and longer side respectively.

Recent studies have concluded that equivalent diameter being derived from empirical relations, is not reliable while calculating pressure losses in pipes. Hence, we use hydraulic diameter in all cases.

## What is the difference between hydraulic diameter, equivalent diameter and characteristic length in fluid mechanics and heat transfer?

Hydraulic diameter, as discussed earlier, is the newly derived diameter from a non-circular duct such that the flow characteristics remain same. Hydraulic diameter is used for calculating Reynold’s number which helps us to understand whether the flow is laminar, transient or turbulent.

The diameter of circular duct which gives same pressure loss as rectangular duct for equal flow is called as equivalent diameter.

Pressure loss in a pipe is given by Darcy-Weisbach equation-

Where,

ρ is the density of the fluid (kg/m^3)

D is the hydraulic diameter of pipe (in m)

l is the length of pipe (in m)

v is the mean flow velocity (in m/s)Characteristic length is basically volume of a system divided by its surface area.

It can be equal to hydraulic diameter in some cases.

Mathematically,

L_{c} = V_{surface}/A_{surface}

For square duct-

L_{c} = a

For rectangular duct-

Lc = 2ab/a+b

In heat transfer, characteristic length is used for calculating Nusselt number.The ratio of convective heat transfer to conductive heat transfer is called as Nusselt number. It shows what type of heat transfer dominates.

Nusselt number, Nu is given by-

N*u* = *h*L*c*/*k*

where,

h is convective heat resistance

L is characteristic length

k is thermal conductivity

Nusselt number of value 1 represents heat transfer by pure conduction, as Nusselt number increases, heat transfer through convection keeps increasing. When the value of Nusselt number approaches 100-1000, convection heat transfer dominates. The value of Nusselt number cannot be less than 1, it can be greater than 1 or equal to 1. Value of Nusselt number is always constant for fully developed laminar flow. For a complex shape, local Nusselt numbers for the surface are calculated and then an average Nusselt number is calculated using these local Nusselt numbers. Average Nusselt number is used for deriving further conclusions.

## What is the difference between hydraulic radius and hydraulic depth / hydraulic mean depth?

There is a misconception that hydraulic radius and hydraulic depth are same. They both have different meanings and hold individual significance while measuring fluid properties. The concept of hydraulic radius and hydraulic depth is discussed in detail below.

The ratio of cross sectional area of flow to the wetted perimeter is called as hydraulic radius.

R_{h} = A/P

The ratio of cross sectional area of flow to free water surface or top surface width is called as hydraulic depth.

H_{d} = A/T

where,

A is the cross sectional area of flow

T is the width up to top surface or free surface.

Mathematically, hydraulic mean depth and hydraulic radius are same.

## What is the physical significance of hydraulic diameter in fluid and thermal sciences?

Practically, Reynold’s number is used to check the behaviour or nature of the fluid flow. This in turn helps us in finding Nusselt number which is then used to find the rate of heat transfer from the closed conduit.

Hence, Reynold’s number is a very important dimensionless number which plays a vital role in both fluid and thermal sciences. But to find Reynold’s number, first we need to find hydraulic diameter of the closed conduit. For non-circular cross sections, hydraulic diameter provides a value of diameter such that its flow characteristics are equivalent to that of a circular cross section.

The ratio of convective heat transfer to conductive heat transfer is called as Nusselt number.

Nusselt number is given by following relation-

For laminar flow: Nu = 0.332 Re^{0.5} Pr^{0.33}

For turbulent flow: Nu = 0.039 Re^{0.8} Pr^{0.33}

Where,

Re denotes Reynold’s number

Pr denotes Prandtl number

The ratio of momentum diffusivity to thermal diffusivity is called as Prandtl number. It is named after German scientist Ludwig Prandtl. This dimensionless number helps us in calculations related to forced and natural heat convection. Its significance is that it helps us to study the relation between momentum transport and thermal transport capacity of fluid.

Prandtl number is calculated by the formula given below-

P*r* = *μ*C*p/k*

Where,

Pr is Prandtl number

µ is dynamic viscosity

Cp is specific heat

Note that Nusselt number can also be found using the relation: Nu = hLc/k, when we know the values of convective and conductive heat resistances.

In simple words, hydraulic diameter forms the basis for finding the behaviour of flow and rate of heat transfer from the fluid that is flowing in a closed conduit. With that, it also brings us easy calculations by converting a non-circular conduit to a circular one.