Combining Frictional and Fitting Pressure Losses

Combining frictional and fitting pressure losses is comparatively straightforward, and we deal with it in much the same way as we do with combined serial fitting losses – we sum the various individual pressure losses. However, it is often necessary, in practice, to be able to determine not only the overall pressure drop along the length of a main or duct but to potentially determine the pressure of the fluid at a specific point along the length. This is often required to ensure that excessively low pressures do not exist in the line at a particular point. This could result in dissolved gases coming out of solution or hot liquid releasing vapours (more on these topics in a later lesson).

The following example illustrates how to determine both overall line pressure drop as well as investigating a particular pressure condition within the line.

A high-gravity beer main is shown below (not to scale) with detail as given. The gauge pressure at inlet to the main (A), is 200 kPa and the flow rate is 400 hL/hr.

What is the overall pressure drop between the inlet (A) and outlet (C)?

Consideration is being given to introduce a dilution stream at point B. In order to ensure that the supply pressure of dilution water is sufficiently high to overcome the beer main pressure, determine what the pressure would be at B.

The pipeline is 75 mm internal diameter with an absolute roughness of 0.05 mm. The length of the bends can be ignored. Beer density is 1,010 kg/m³ and dynamic viscosity is 3 cP.

calculation

As we will require the average flow velocity to determine velocity heads for the fitting losses, and the flow Reynolds number for pipe friction factor, let us work these out as we have done before:

Q equals A u

So, rearranging for u:

u equals Q over A equals fraction numerator begin display style fraction numerator begin display style fraction numerator 400 cross times 100 over denominator 3 comma 600 cross times 1 comma 000 end fraction end style over denominator pi cross times 0.075 to the power of ² end fraction end style over denominator 4 end fraction equals 2.5 m divided by s

Hence the pipe Re is:

R e equals fraction numerator rho u d over denominator mu end fraction equals fraction numerator 1 comma 010 cross times 2.5 cross times 0.075 over denominator 3 cross times 10 to the power of negative ³ end exponent end fraction equals 6.3 cross times 10 to the power of ⁴

Let us deal with the friction pressure loss first for the entire pipe run. The Darcy-Weisbach equation is:

increment P subscript բ equals 4 phi L over d rho u to the power of ²

The complete pipe length L is (10 + 5 + 5 + 10 + 5 + 5) = 40 m.

In order to determine the friction factor from a Moody diagram, we need the Re (which we have just calculated) and the relative roughness ε, which is given by:

epsilon equals e over d equals fraction numerator 0.05 over denominator 75 end fraction equals 0.0007

From the Moody diagram, the friction factor φ is 0.0054. 

pipe friction chart

Therefore, the total friction pressure drop is given by:

increment P subscript բ equals 4 cross times 0.0054 cross times fraction numerator 40 over denominator 0.075 end fraction cross times 1 comma 010 cross times 2.5 to the power of ²

increment P subscript բ equals 72 comma 720 space P a space o r space 72.7 space k P a

Let us also express this as a friction pressure drop per metre run of pipe, (72.7/40) = 1.82 kPa/m.

The total fitting loss will be the sum of the loss coefficients for the four bends and the open globe valve:

increment P subscript f i t t i n g end subscript equals left parenthesis fraction numerator u to the power of ² rho over denominator 2 end fraction right parenthesis space left square bracket 2 cross times 1.8 plus 2 cross times 0.9 plus 10 right square bracket

increment P subscript f i t t i n g end subscript equals left parenthesis fraction numerator 2.5 to the power of ² cross times 1 comma 010 over denominator 2 end fraction right parenthesis space left square bracket 15.4 right square bracket

increment P subscript f i t t i n g end subscript equals 48 comma 606 space P a space o r space 48.6 space k P a

Therefore, the total pipeline pressure drop is given (72.7 + 48.6) = 121.3 kPa.

In order to determine the pressure at B, we need to determine the pressure drop between A and B, which consists of a pipe run of (10 + 5 + 5) = 20 m and a 180° bend and an open globe valve.

We previously determined that the friction loss per metre of pipe run is 1.82 kPa/m, hence the friction pressure drop of the 20 m pipe run is 20 x 1.82 = 36.4 kPa.

The pressure loss due to the fittings is given by:

increment P subscript f i t t i n g end subscript equals left parenthesis fraction numerator u to the power of ² rho over denominator 2 end fraction right parenthesis space left square bracket 1.8 plus 10 right square bracket

increment P subscript f i t t i n g end subscript equals left parenthesis fraction numerator 2.5 to the power of ² cross times 1010 over denominator 2 end fraction right parenthesis space left square bracket 11.8 right square bracket

increment P subscript f i t t i n g end subscript equals 37.2 k P a

Hence ∆PA-B = 73.6 kPa

It is worth noting that the fitting and friction pressure losses are very similar, which may be unexpected and is largely due to the high loss coefficient of the open globe valve. This is an example of taking note of high loss coefficient fittings, and in practice, questioning whether they cannot be replaced by lower loss coefficient alternatives, or moved to a different part of the pipeline.

As PA = 200 kPa. We can determine PB:

increment P subscript A minus B end subscript equals P subscript A minus P subscript B equals 73.6 space k P a

P subscript B equals P subscript A minus increment P subscript A minus B end subscript equals 200 minus 73.6

P subscript B equals 126.4 space k P a

We have now dealt with pressure drops in the pipe/duct system, and the different causes of pressure drop. We will now finish this lesson with a few thoughts on how we choose flow velocities and allowable pressure drops.

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