Hydraulic Mean Depth Hydraulic Radius Hydraulic Diameter PDF

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Hydraulic Mean Depth, Hydraulic Radius and Hydraulic Diameter Chapter · November 2019

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A. Hydraulic Mean Depth, Hydraulic Radius and Hydraulic Diameter i. Hydraulic Mean Depth, Hydraulic Radius 1. Definition formula Hydraulic Mean Depth (h) ℎ=

𝐴 𝑇

Where,

A = Cross section Area T = Top Width

Hydraulic Radius (R) 𝑅=

𝐴 𝑃

Where,

A = Cross section Area P = Wetted perimeter

Parameters of h and R can be seen in the figure 1 also. Fig. 1. Irregular Cross Section

Fig. 2 Equivalent sections of Irregular Cross sections (fig. a) in terms of Hydraulic mean depth (fig. b) and Hydraulic radius (fig. c)

Fig. 2 (a) shows irregular cross section with Area ‘A’, Top width ‘T’ and wetted Perimeter ‘P’. Then transform this irregular section into a rectangular section such that Area is unchanged. In hydraulic mean depth, top width ‘T’ is used, and since the area is to remain unchanged, depth of the rectangular section (h) is put such that h * T = A as shown in fig. 2 (b). In hydraulic radius, wetted perimeter ‘P’ is used, and since the area is to remain unchanged, depth of the rectangular section (R) is put such that R * P = A as shown in fig. 2 (c).

2. Where are they used? Hydraulic Mean Depth (h) a. It is used in Froude number (

𝑉 √𝑔ℎ

)and energy relationship in open channel flow

Hydraulic Radius (R) a. While calculating frictional head losses, Hydraulic radius is used. While deriving Darcy-Weisbach equation for frictional loss, we get a term, A/P (= R) and for full flowing circular pipe (diameter d), we write d/4 i.e. 𝜋𝑑 2 /4 𝐴 𝑑 = 𝑅= = 𝑃 4 𝜋. 𝑑

3. What happens in rectangular channel? Hydraulic Mean Depth (h) a. h is equal to depth of rectangular flow section. Let Rectangular channel have width B and depth of flow y then 𝐴 𝐵. 𝑦 = 𝑦 ℎ= = 𝐵 𝑇

Hydraulic Radius (R) a. R isn’t equal to depth in rectangular channel but becomes nearly equal to depth of flow as the channel becomes wide. 𝐵. 𝑦 𝐴 = 𝑅= 𝑃 𝐵 + 2𝑦 When channel is wide i.e. B + 2y becomes B then 𝐴 𝐵. 𝑦 𝐵. 𝑦 = 𝑦 𝑅= = = 𝐵 𝑃 𝐵 + 2𝑦

4. Significance of Hydraulic Radius 𝐴 𝑃 If R is higher, P will be lower. It means less amount of water is in contact with the channel section and hence, resistance to the flow will be less letting more discharge to pass through it. Thus, higher R gives better efficiency.

𝑅=

Rectangular Channel ii.

Hydraulic Diameter Hydraulic Diameter Definition For complicated geometries, Hydraulic diameter (HD) is used and is given byHD = 4 * A/P = 4 * R It converts cross-section of any shape into circular cross-section (hydraulically more or less).

Hydraulic Diameter Vs Geometric Diameter Hydraulic radius is different from geometric radius. Geometric diameter is twice the geometric radius but Hydraulic diameter is four times the Hydraulic radius.

Hydraulic Diameter of Circle and Square For full flowing circular section (diameter d), fortunately, diameter and Hydraulic diameter are same. Circle

𝑑 𝜋𝑑 2 /4 𝐴 = = 𝑅= 𝜋. 𝑑 𝑃 4 𝑑 And, Hydraulic Diameter (HD) = 4 * HR = 4 ∗ = d, which is the geometric diameter of 4 circle. Similarly for full flowing square section with side ‘a’, we get hydraulic radius as a/4 𝐴 𝑎. 𝑎 𝑎 𝑅= = = 4 𝑃 4. 𝑎 Square

And, Hydraulic Diameter (HD) = 4 * HR = 4 ∗

𝑎 4

= a, which is the side of the square.

Hydraulic Diameter and Hydraulic radius of different shapes of closed and open channel i. Rectangle (closed channel) Hydraulic radius for rectangle B Y A 5 3 15 HR 0.9375 HD = 4 * 0.9375 = 3.75

P 16

Hydraulic radius for converted circle HD A P HR 𝐴=

𝜋.(𝐻𝐷)2 4

3.75 11.04466 11.78097 0.9375

HD = 3.75

𝑃 = 𝜋. 𝐻𝐷

Hydraulic radius for subtracted section [abs (Rectangle – Circle)] ΔA ΔP

3.955338 4.219028

HR 0.9375

When we subtract Area of rectangle and circle (say ΔA) and perimeter of rectangle and circle (say ΔP) and find Hydraulic radius as∆𝐴 𝑅= ∆𝑃 We will get same hydraulic radius. This can be seen highlighted in green color in the above three tables.

ii.

Rectangle (Open Channel) Hydraulic radius for rectangle B Y A 5 2 10 HR 1.111111 HD = 4 * 1.111111 = 4.444444

P 9

Hydraulic radius for converted circle HD A P HR 𝐴=

𝜋.(𝐻𝐷)2 4

4.444444 15.51404 13.96263 1.111111

HD = 4.444444

𝑃 = 𝜋. 𝐻𝐷

Hydraulic radius for subtracted section [abs (Rectangle – Circle)] ΔA ΔP

5.514038 HR 4.962634 1.111111

When we subtract Area of rectangle and circle (say ΔA) and perimeter of rectangle and circle (say ΔP) and find Hydraulic radius as∆𝐴 𝑅= ∆𝑃 We will get same hydraulic radius. This can be seen highlighted in green color in the above three tables.

iii.

Trapezoid (Open Channel) Hydraulic radius for Trapezoid B Y Z A=By+ZY^2 5 3 1 24 HR 1.779718 1V:ZH HD = 4 * 1.779718= 7.118873

P=B+2Y*sqrt(z^2+1) 13.48528137

Hydraulic radius for converted circle HD A P HR 𝐴=

𝜋.(𝐻𝐷)2 4

7.118873 39.80268 22.3646 1.779718

HD = 7.118873

𝑃 = 𝜋. 𝐻𝐷

Hydraulic radius for subtracted section [abs (Trapezoid – Circle)] ΔA ΔP

15.80268 HR 8.879316 1.779718

When we subtract Area of trapezoid and circle (say ΔA) and perimeter of trapezoid and circle (say ΔP) and find Hydraulic radius as-

∆𝐴 ∆𝑃 We will get same hydraulic radius. This can be seen highlighted in green color in the above three tables. The same result will be yielded with the full flow trapezoidal section. 𝑅=

iv.

Semi-Circle (Open Channel) Hydraulic radius for semi-circle r D 2.5 5 HR 1.25 HD = 4 * 1.25= 5

A 9.817477

P 7.853982

Hydraulic radius for converted circle HD A P HR 𝐴=

5 19.63495 15.70796 1.25 𝜋.(𝐻𝐷)2 4

HD = 5

𝑃 = 𝜋. 𝐻𝐷

Hydraulic radius for subtracted section [abs (semi-circle – Circle)] ΔA ΔP

9.817477 HR 7.853982 1.25

When we subtract Area of semi-circle and circle (say ΔA) and perimeter of semicircle and circle (say ΔP) and find Hydraulic radius as∆𝐴 𝑅= ∆𝑃 We will get same hydraulic radius. This can be seen highlighted in green color in the above three tables. The same result will be yielded with the full flow of semicircle section....