Pipe Friction, Reynolds number & Bernoilli's Equation

[sci0x]

Homework Statement

A pump is used to empty a fermenting vessel into a maturation vessel. Fermenting vessel is open to atmps pressure. Maturation vessel has 100 kPa pressure maintained. On transfer completion, maturation vessel is filled to 25m depth.

Using Pipe Friction chart, and assuming smooth pipes, calculate the friction factor for losses in the pipework

Data: pipe inside diameter = 100mm
Connecting pipe length = 50m
Distance of pump below fermentor vessel = 1.5m
Distance of pump below maturation vessel = 2.0m
Mean flow velocity along pipe = 1.5ms-1
Pump Effeciency = 60%
Density of beer = 1007 kgm-3
Beer Viscosity = 0.002 Pa s
Acc due to grav = 9.81ms-2

Relevant Equations

Reynolds no. Re

First calc reynolds no. Re

Flow = 1.5ms-2

Re = (density x mean velocity x diameter) / viscosity
= (1007 kgm-3)(1.5ms-1)(0.1m) / 0.002 Pa s
= 75,525 = Turbulent flow

I used this chart Here
And I am getting friction factor of 0.02
Would you agree with this?

 

[Chestermiller]

It looks to me more like 0.019. But, approach is correct.

 

[sci0x]

Great Chester.

The next Q is: Set out the Bernouilli energy balance equation which describes the system at maximum pump power demand:

My answer: Atmospheric pressure + Depth of beer to Transfer + Distance of pump below fermenting vessel + Pump Power = Pressure on maturation vessel + Pressure drop due to Friction + Distance of pump below maturation vessel

Atmospheric pressure = 101.3kPa = 101300 Pa
Convert to m: 101300 Pa / (1007 kgm-3)(9.81ms-2) = 10.25m

Depth of beer to transfer = 25m

Distance of pump below fermenting vessel = 1.5m

Pump power = x

Pressure on maturation vessel = 100kPa = 100,000 Pa
100,000 Pa / (1007 kgm-3)(9.81 ms-2) = 10.12m

Pressure drop due to Friction:
(4)(friction factor)(density)(pipe length)(mean velocity)^2 / pipe diameter
(4)(0.019)(50m)(1007kgm-3)(1.5m/s)^2 / 0.1m = 86,098.5 Pa
86,098.5 / (1007 kgm-3)(9.81ms-2) = 8.72m

Distance of pump below maturation vessel = 2.0m

Bernouilli Eq: 10.25m + 25m + 1.5m + Pump Power= 10.12m + 8.72m + 2.0m

Im assuming here that the depth of the beer is also 25m in the fermenting vessel.
Let me know if you agree with this?

 

[Chestermiller]

I'm having trouble visualizing the arrangement. Can you please provide a schematic with the key distances and dimensions shown on the figure?

 

[sci0x]

Hi Chester, this is my understanding of the Question

My problem is that I am ending up with negative pump power

 

[Chestermiller]

Your pressure drop for flow is high by a factor of 8. There should be a 1/2 out front, not a 4. So, it's 10.76 kPa.

I assume that the pressure in the maturation vessel is 100 kPa gauge. So the actual pressure is 201.3 kPa absolute. Using Bernoulli, $$101.3+9.9+\Delta p=201.3+266.7+10.8$$where $\Delta p$ is the pressure increment provided by the pump. This gives $$\Delta p=166.3\ kPa$$This assumes that the height of the fluid in the fermentation vessel is zero at the end. The power required from the pump is the pressure increment provided by the pump multiplied by the volumetric flow rate 0.0118 m^3/s: W = 1.96 kW. With a 60% efficiency, the actual power should be 3.27 kW.

 

[sci0x]

Can i ask where you got the the 9.9 and 266.7 in the Bernouilli Eq

Also in the exam Q, they gave Darcys Eq, and use 4, not 1/2.

Exam notes say answer for pump power is 7.1 kW

 

[Chestermiller]

Can i ask where you got the the 9.9 and 266.7 in the Bernouilli Eq

$$\rho g h=(1.007)(9.81)(1.5)=14.8\ kPa$$
my mistake. The 9.9 should be 14.8

$$\rho g h=(1.007)(9.81)(2+25)=266.7\ kPa$$
It looks like I did the arithmetic wrong to get $\Delta p$. It should be $$\Delta P=266.7+201.3+10.8-101.3-14.8=362.7\ kPa$$
So the pump pressure boost should be 362.7 kPa

Also in the exam Q, they gave Darcys Eq, and use 4, not 1/2.

Check the equation given on the bottom of your Moody diagram (which is the correct equation)

Exam notes say answer for pump power is 7.1 kW

What do you get for the volume flow rate? What do you get for the pump pressure boost times the volume flow rate in kW? What do you get when you divide this by 0.6?