Friction in Pipe: Understanding N in Head Loss Formula

[foo9008]

Homework Statement

what is the meaning of n here ? i only know the formula of head loss is f(L/ D ) (V^2) / (2g)

Homework Equations

The Attempt at a Solution

 

[SteamKing]

Homework Statement

what is the meaning of n here ? i only know the formula of head loss is f(L/ D ) (V^2) / (2g)

Homework Equations

The Attempt at a Solution

Apparently, n is the exponent of the flow velocity V as shown on the diagram.

For laminar flow, the friction factor f is directly proportional to V.

For the transition zone, the friction factor is proportional to a certain power of V, which is expressed as f ∝ Vⁿ, where n = 1.75 - 2.0.

For fully turbulent flow, f ∝ V².

 

[foo9008]

Apparently, n is the exponent of the flow velocity V as shown on the diagram.

For laminar flow, the friction factor f is directly proportional to V.

For the transition zone, the friction factor is proportional to a certain power of V, which is expressed as f ∝ Vⁿ, where n = 1.75 - 2.0.

For fully turbulent flow, f ∝ V².

so , the formula of head loss still f (L)(v^n) / (2gD ) ? at turbulent region ?

 

[SteamKing]

so , the formula of head loss still f (L)(v^n) / (2gD ) ? at turbulent region ?

Yes, if you make n = 2.

 

[foo9008]

Apparently, n is the exponent of the flow velocity V as shown on the diagram.

For laminar flow, the friction factor f is directly proportional to V.

For the transition zone, the friction factor is proportional to a certain power of V, which is expressed as f ∝ Vⁿ, where n = 1.75 - 2.0.

For fully turbulent flow, f ∝ V².

the book gave that for fully turbulent flow , n = 1.75-2.0 , THE CORRECT SHOULD BE for transition zone flow , n = 1.75-2.0 ?

 

[SteamKing]

the book gave that for fully turbulent flow , n = 1.75-2.0 , THE CORRECT SHOULD BE for transition zone flow , n = 1.75-2.0 ?

The roughness of the pipe wall also has an influence.

Look, the book is trying to establish an approximate relationship between the velocity of the flow and the friction factor, as illustrated by the diagram. For whatever reason, they would like this relationship to be a smooth curve throughout the flow regimes from purely laminar flow to purely turbulent flow.

 

[foo9008]

The roughness of the pipe wall also has an influence.

Look, the book is trying to establish an approximate relationship between the velocity of the flow and the friction factor, as illustrated by the diagram. For whatever reason, they would like this relationship to be a smooth curve throughout the flow regimes from purely laminar flow to purely turbulent flow.

for transition region , the relationship between the velocity and friction factor is not constant , it may vary ... so there's is discontinuous line on the graph , am i right ?

 

[SteamKing]

for transition region , the relationship between the velocity and friction factor is not constant , it may vary ... so there's is discontinuous line on the graph , am i right ?

The dashed line in the transition region appears to be an extrapolation of what happens after fully laminar flow has stopped and when fully turbulent flow is established.
When flow is laminar, the friction factor is directly proportional to flow velocity; when flow is fully turbulent, friction factor is proportional to flow velocity raised to the power indicated by the exponent on the graph. In the transition zone, there is some non-linear relationship between friction factor and flow velocity; it's not entirely linear, but it's not fully quadratic, either.

 

[foo9008]

The dashed line in the transition region appears to be an extrapolation of what happens after fully laminar flow has stopped and when fully turbulent flow is established.
When flow is laminar, the friction factor is directly proportional to flow velocity; when flow is fully turbulent, friction factor is proportional to flow velocity raised to the power indicated by the exponent on the graph. In the transition zone, there is some non-linear relationship between friction factor and flow velocity; it's not entirely linear, but it's not fully quadratic, either.

The dashed line in the transition region appears to be an extrapolation of what happens after fully laminar flow has stopped and when fully turbulent flow is established.
When flow is laminar, the friction factor is directly proportional to flow velocity; when flow is fully turbulent, friction factor is proportional to flow velocity raised to the power indicated by the exponent on the graph. In the transition zone, there is some non-linear relationship between friction factor and flow velocity; it's not entirely linear, but it's not fully quadratic, either.

Darcy -weisbech equationgave that hf = fL(v^2) / 2gD , and Darcy -weisbech equation is for both laminar and turbulent , am i right ? why according to the graph , the friction factor is directly proportional to hf / shouldn't it be n= 2 ?

 

[SteamKing]

Darcy -weisbech equationgave that hf = fL(v^2) / 2gD , and Darcy -weisbech equation is for both laminar and turbulent , am i right ? why according to the graph , the friction factor is directly proportional to hf / shouldn't it be n= 2 ?

If you look at the formulations of the D-W friction factor, you'll see that f for laminar flow is very different for f for fully turbulent flow.

https://en.wikipedia.org/wiki/Darcy–Weisbach_equation

You also shouldn't confuse how f behaves with flow velocity with how head loss behaves with flow velocity. The Moody diagram, for example, establishes that for turbulent flow, f is independent of flow velocity.

The diagram in the OP documents Reynold's measurements of friction versus pipe length. It's not clear how well Reynold's measurements correlate with the work of others based on this one diagram.

 

[foo9008]

If you look at the formulations of the D-W friction factor, you'll see that f for laminar flow is very different for f for fully turbulent flow.

https://en.wikipedia.org/wiki/Darcy–Weisbach_equation

You also shouldn't confuse how f behaves with flow velocity with how head loss behaves with flow velocity. The Moody diagram, for example, establishes that for turbulent flow, f is independent of flow velocity.

The diagram in the OP documents Reynold's measurements of friction versus pipe length. It's not clear how well Reynold's measurements correlate with the work of others based on this one diagram.

i know that , when the flow is fully turbulent , the friction factor is independent of velocity , but for both laminar and fully turbulent flow , the head loss has the formula of fL(v^2) / 2gD , so for both flow , the hf is directly proportional to v^2 , am i right ? but the diagram for v and hf is proportional, is the diagram wrong ?
P/s: in the first diagram , it's a graph if hf vs velocity , not friction factor vs velocity

 

[SteamKing]

i know that , when the flow is fully turbulent , the friction factor is independent of velocity , but for both laminar and fully turbulent flow , the head loss has the formula of fL(v^2) / 2gD , so for both flow , the hf is directly proportional to v^2 , am i right ? but the diagram for v and hf is proportional, is the diagram wrong ?
P/s: in the first diagram , it's a graph if hf vs velocity , not friction factor vs velocity

There's only one diagram: a plot of head loss versus flow velocity made using data recorded by Reynolds.

Apparently, Reynolds chose not to use the modern formula, HL = f (L/D) v² / 2g to plot his results. That's why there's a range of exponents. IDK why Reynolds chose this method; if you want to know, you'll have to research Reynolds work on the matter.

 

[foo9008]

There's only one diagram: a plot of head loss versus flow velocity made using data recorded by Reynolds.

Apparently, Reynolds chose not to use the modern formula, HL = f (L/D) v² / 2g to plot his results. That's why there's a range of exponents. IDK why Reynolds chose this method; if you want to know, you'll have to research Reynolds work on the matter.

So, this is for to show expermintal data only? In real life, we wouldq use the formula hf = f(L/D)(V^2)/2g ?