# Head loss in pipe

## Homework Statement

estimate the energy head lost along a short length of pipe suddenly enlarging a diameter of 350mm to 700mm which discharges 0.7(m^-3) of water per second .
the solution given is Q1= Q2 = 0.7(m^3)/s
0.7 = pi ((350x10^-3)^2 ) V1 / 4 , V1= 7.28m/s
0.7 = pi ((700x10^-3)^2 ) V2 / 4 , v2 = 1.82m/s
[(v1)^2 ] / 2g = [(v2)^2 ] / 2g +head loss
head loss = 2.53m

## The Attempt at a Solution

the bernoulli's equation is P1/ ρg + (V1^2) /2g + z1 = P2/ ρg + (V2^2) /2g + z2, by writing [(v1)^2 ] / 2g = [(v2)^2 ] / 2g +head loss , the author assume P1 = P2 , how could it be possible ? since the inlet is smaller than outlet , the pressure at inlet should be lower , right ? the velocity at inlet is smaller

## Answers and Replies

Chestermiller
Mentor
The head loss is ##\frac{(P_1-P_2)}{\rho g}##

The head loss is ##\frac{(P_1-P_2)}{\rho g}##
but i dont have P1 and P2.... how to solve this ?

Chestermiller
Mentor
but i dont have P1 and P2.... how to solve this ?
You're solving algebraically for the entire quantity ##\frac{(P_1-P_2)}{\rho g}## (which is the head loss), without having to know either P1 or P2 individually. You are aware that the pipe is assumed to be horizontal so that z1 = z2, correct? And you are aware of the definition of head, correct?

• foo9008
You're solving algebraically for the entire quantity ##\frac{(P_1-P_2)}{\rho g}## (which is the head loss), without having to know either P1 or P2 individually. You are aware that the pipe is assumed to be horizontal so that z1 = z2, correct? And you are aware of the definition of head, correct?
yes , [(v1)^2 ] / 2g = [(v2)^2 ] / 2g +head loss is it correct ??

Chestermiller
Mentor
yes , [(v1)^2 ] / 2g = [(v2)^2 ] / 2g +head loss is it correct ??
No. There should be a minus sign in front of head loss.

No. There should be a minus sign in front of head loss.
the energy flow from point 1 to 2 , so we should add the head loss at point 2 , right ?

Chestermiller
Mentor
The head loss is defined as the decrease in head from the upstream point to the downstream point.

• foo9008
The head loss is defined as the decrease in head from the upstream point to the downstream point.
ya , point 1 is upstream , point 2 is downstream , right ?

The head loss is ##\frac{(P_1-P_2)}{\rho g}##
why the head loss is ##\frac{(P_1-P_2)}{\rho g}##?
shouldn't it be P1/ ρg + (V1^2) /2g + z1 = P2/ ρg + (V2^2) /2g + z2 + head loss ???? we cancel our z1 and z2 , since they are the same

Chestermiller
Mentor
why the head loss is ##\frac{(P_1-P_2)}{\rho g}##?
shouldn't it be P1/ ρg + (V1^2) /2g + z1 = P2/ ρg + (V2^2) /2g + z2 + head loss ???? we cancel our z1 and z2 , since they are the same
What is your understanding of the term head, and what is your understanding of the term head loss?

• foo9008
What is your understanding of the term head, and what is your understanding of the term head loss?
A portion of energy is lost to the resistance to flow....

Chestermiller
Mentor
A portion of energy is lost to the resistance to flow....
That is not the general definition of head loss and, in this problem, the resistance to flow is implicitly assumed to be zero. More generally, head loss is the decrease in ##P/(\rho g)+z## between inlet and outlet.

That is not the general definition of head loss and, in this problem, the resistance to flow is implicitly assumed to be zero. More generally, head loss is the decrease in ##P/(\rho g)+z## between inlet and outlet.
can you help how to solve this question ?

That is not the general definition of head loss and, in this problem, the resistance to flow is implicitly assumed to be zero. More generally, head loss is the decrease in ##P/(\rho g)+z## between inlet and outlet.
sorry , i missed out part of thge question . the full question given is
estimate the energy head lost along a short length of pipe suddenly enlarging a diameter of 350mm to 700mm which discharges 0.7(m^-3) of water per second .. If the pressure at entrance is 105N/ (m^2) . Find the pressure at exit

That is not the general definition of head loss and, in this problem, the resistance to flow is implicitly assumed to be zero. More generally, head loss is the decrease in ##P/(\rho g)+z## between inlet and outlet.
so the equation should be (105x10^3)/ (1000x9.81) + (7.28^2) / (2x9.81) +z1 = P2 / (1000x9.81) + (1.82^2) / (2x9.81) + head loss , am i right ?
cancel off z1 and z2 , we still have unknown P2 , how to find the head loss?

Chestermiller
Mentor
so the equation should be (105x10^3)/ (1000x9.81) + (7.28^2) / (2x9.81) +z1 = P2 / (1000x9.81) + (1.82^2) / (2x9.81) + head loss , am i right ?
cancel off z1 and z2 , we still have unknown P2 , how to find the head loss?
No. It's not correct.

$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$

• foo9008
No. It's not correct.

$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$
pls refer to the link diagram 6.5 , the formula fo hf is : http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Examples/eg6_ans.htm

No. It's not correct.

$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$ No. It's not correct.

$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$
View attachment 99132

No. It's not correct.

$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$
is it because when we find the head loss , we pick 2 points which have the same velocity and same pressure.... so we can have only P1 and P2 (2 unknowns) ?
in my calculation , $$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$ , i have chose the wrong points , which have different pressure and different velocity , so i am having problems of finding the head loss?

Chestermiller
Mentor
pls refer to the link diagram 6.5 , the formula fo hf is : http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Examples/eg6_ans.htm
In this equation, hf is the frictional head loss, not the total head loss. The total head loss is h1-h2, where $$h=\frac{p}{\rho g}+z$$

• foo9008
Chestermiller
Mentor
is it because when we find the head loss , we pick 2 points which have the same velocity and same pressure.... so we can have only P1 and P2 (2 unknowns) ?
No. We pick points at the inlet and outlet.
in my calculation , $$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$ , i have chose the wrong points , which have different pressure and different velocity , so i am having problems of finding the head loss?
No. This equation is correct. To get the head loss, you don't have to know P1 and P2 individually. You just have to determine their difference. Do you know how to do the algebra to get their difference?

• foo9008
No. We pick points at the inlet and outlet.

No. This equation is correct. To get the head loss, you don't have to know P1 and P2 individually. You just have to determine their difference. Do you know how to do the algebra to get their difference?
as you said in the previous post , The head loss is ##\frac{(P_1-P_2)}{\rho g}## ? so ,
$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$ ??

Chestermiller
Mentor
as you said in the previous post , The head loss is ##\frac{(P_1-P_2)}{\rho g}## ? so ,
$$\frac{P_1}{\rho g}+z_1+\frac{(7.28)^2}{(2)(9.81)}=\frac{P_2}{\rho g}+z_2+\frac{(1.82)^2}{(2)(9.81)}$$
$$\frac{P_1}{\rho g}-\frac{P_2}{\rho g}=head\ loss=\frac{(1.82)^2}{(2)(9.81)}-\frac{(7.28)^2}{(2)(9.81)}=-2.53m$$So,
$$\frac{P_2}{\rho g}=\frac{P_1}{\rho g}+2.53m$$ ??
Is there a question here?

• foo9008