Minor Head Loss – Local Losses
In industry any pipe system contains different technological elements as bends, fittings, valves or heated channels. These additional components add to the overall head loss of the system. Such losses are generally termed minor losses, although they often account for a major portion of the head loss. For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses (especially with a partially closed valve that can cause a greater pressure loss than a long pipe, in fact when a valve is closed or nearly closed, the minor loss is infinite).
The minor losses are commonly measured experimentally. The data, especially for valves, are somewhat dependent upon the particular manufacturer’s design.
Like pipe friction, the minor losses are roughly proportional to the square of the flow rate and therefore they can be easy integrated into the Darcy-Weisbach equation. K is the sum of all of the loss coefficients in the length of pipe, each contributing to the overall head loss.
There are several methods how to calculate head loss from fittings, bends and elbows. In the following section these methods are summarized in the order from the simplest to the most sophisticated.
Equivalent Length Method
The equivalent length method (The Le/D method) allows the user to describe the pressure loss through an elbow or a fitting as a length of straight pipe.
This method is based on the observation that the major losses are also proportional to the velocity head (v2/2g).
The Le/D method simply increases the multiplying factor in the Darcy-Weisbach equation (i.e. ƒ.L/D) by a length of straight pipe (i.e. Le) which would give rise to a pressure loss equivalent to the losses in the fittings, hence the name “equivalent length”. The multiplying factor therefore becomes ƒ(L+Le)/D and the equation for calculation of pressure loss of the system is therefore:
All fittings, elbows, tees, can be summed up to make one total length, and the pressure loss calculated from this length. It has been found experimentally that if the equivalent lengths for a range of sizes of a given type of fitting are divided by the diameters of the fittings then an almost constant ratio (i.e. Le/D) is obtained. The advantage of the equivalent length method is that a single data value is sufficient to cover all sizes of that fitting and therefore the tabulation of equivalent length data is relatively easy. Some typical equivalent lengths are shown in the table.
Resistance Coefficient Method – K Method – Excess head
The resistance coefficient method (or K-method, or Excess head method) allows the user to describe the pressure loss through an elbow or a fitting by a dimensionless number – K. This dimensionless number (K) can be incorporated into the Darcy-Weisbach equation in a very similar way to the equivalent length method. Instead of of equivalent length data in this case the dimensionless number (K) is used to characterise the fitting without linking it to the properties of the pipe.
The K-value represents the multiple of velocity heads that will be lost by fluid passing through the fitting. The equation for calculation of pressure loss of the hydraulic element is therefore:
Therefore the equation for calculation of pressure loss of entire hydraulic system is:
The K-value can be characterised for various flow regime (i.e. according to the Reynolds number) and this causes it is more accurate than the equivalent length method.
There are several other methods for calculating pressure loss for fittings, these methods are more sophisticated and also more accurate:
- 2K-Method. The 2K method is a technique developed by Hooper B.W. to predict the head loss in an elbow, valve or tee. The 2K method improves on the excess head method by characterising the change in pressure loss due to varying Reynolds number. The 2-K method is advantageous over other method especially in the laminar flow region.
- 3K-Method. The 3K method (by Ron Darby in 1999) further improves the accuracy of the pressure loss calculation by also characterising the change in geometric proportions of a fitting as its size changes. This makes the 3K method particularly accurate for a system with large fittings.
Flow through Elbow – Minor Loss
The flow through elbows is quite complicated. In fact, any curved pipe always induces a larger loss than the simple straight pipe. This is due to the fact in a curved pipe the flow separates on the curved walls. For very small radius of curvature the incoming flow is even unable to make the turn at the bend, therefore the flow separates and in part stagnates against the opposite side of the pipe. In this part of the bend the pressure raises (as a result of the Bernoulli’s principle) and the velocity decreases.
An interesting feature of the K-values for elbows is their non-monotone behavior as R/D ratio increases. The K-values include both the local losses and frictional losses of the pipe. The local losses, caused by flow separation and secondary flow, decrease with R/D, while the frictional losses increase because the bend length increases. Therefore there is a minimum in the K-value near the normalized radius of curvature of 3.
It is known that when the fluid is heated (e.g. in a fuel channel), the fluid expands (change in the fluid density) and increases its flow velocity as a result of the continuity equation (the channel cross-section remains the same). For a control volume that has a single inlet and a single outlet, this equation states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out.
See also: Subcooled Water Properties
Another very important principle states (Bernoulli’s principle) that the increase in flow velocity in the heated channel causes the lowering of fluid pressure. This pressure loss can be also considered as a local pressure loss and can be calculated from the following equation:
Flow rate through a reactor core – coolant acceleration
It is an illustrative example, following data do not correspond to any reactor design.
Pressurized water reactors are cooled and moderated by high-pressure liquid water (e.g. 16MPa). At this pressure water boils at approximately 350°C (662°F). Inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m3). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m3) as the water flows through the core.
The primary circuit of typical PWRs is divided into 4 independent loops (piping diameter ~ 700mm), each loop comprises a steam generator and one main coolant pump. Inside the reactor pressure vessel (RPV), the coolant first flows down outside the reactor core (through the downcomer). From the bottom of the pressure vessel, the flow is reversed up through the core, where the coolant temperature increases as it passes through the fuel rods and the assemblies formed by them.
- Pressure loss due to the coolant acceleration in an isolated fuel channel
- channel inlet flow velocity is equal to 5.17 m/s
- channel outlet flow velocity is equal to 5.69 m/s
The pressure loss due to the coolant acceleration in an isolated fuel channel is then:
This fact has important consequences. Due to the different relative power of fuel assemblies in a core, these fuel assemblies have different hydraulic resistance and this may induce local lateral flow of primary coolant and it must be considered in thermal-hydraulic calculations.