In order to evaluate the need for circulation pump capacity, a series of calculations related to the fluid flow is needed. These calculations involve the length of the pipes carrying the fluid, the optimal diameter of the pipe and other factors. These calculations will in the end determine the pressure drop per meter, thus dictate proper circulation pump capacity.
Circular pipes are able to withstand high pressure differences between the inside and outside, and is therefore often used for transportation of liquids.
2.4.1 Dynamic Viscosity
Dynamic viscosity,µ[kg/m s], is an expression for internal resistance within a fluid, saying something about the horizontal movement of the fluid within. The viscosity is dependent on the temperature of the fluid, and is calculated differently for liquids and gases.
µ= aT1/2
1 +b/T [kg/m s] (2.10)
Equation 2.10 is the dynamic viscosity for gases, and the constants a and b are found experimentally, whileT is the absolute temperature. For water at atmospheric conditions,a and b are approximated to be as follows:
• a = 1.458·10−6 kg/m s K1/2
• b = 110.4K
As indicated by equation 2.10, the viscosity increases with the temperature.
Equation 2.11 is used to calculate the dynamic viscosity of liquids.
µ=a·10b/(T−c) [kg/m s] (2.11) The constantsa,bandcare found experimentally. For water, using the following values:
Knowingµ, the Reynolds number, Re [−], can be found, using equation 2.12. The Reynolds number indicates how the fluid flows inside the pipe. The flow is either categorized as laminar or turbulent, depending on Re being above or below 2300.
Below 2300 is laminar, and above it is considered turbulent. However, it is not fully turbulent before reaching 4000, thus between 2300 and 4000 the flow is described as transitional. [4]
A laminar flow is characterized by having smooth and ordered lines. Turbulent flows are recognized by unsteadiness and wavering lines. The transitional flow is the transition between laminar and turbulent flows. Laminar flows are mostly present in fluids with high viscosity, like oils, or through narrow pipes. Most flows for heat transfer are therefore turbulent.
Re = Inertial forces
Viscous forces = ρvavgD
µ [−] (2.12)
ρ[kg/m3] is the density of the liquid. vavg [m/s] is the velocity of the liquid, and D[m] is the diameter of the pipe.
In a pipe, the velocity varies greatly through the cross section. At the pipe wall, the velocity is equal to zero, and it increases towards the center of the pipe.
It is quite difficult to perform calculations for this, and therefore vavg makes the calculations easier. vavg is the average velocity through the whole pipe. An average temperature T is also used. The losses in accuracy caused by these simplifications are minor to the increased convenience.
When the Reynolds number is high, the inertial forces are superior to the viscous forces, meaning the viscous effect cannot prevent the fluctuations happening. For smaller Reynolds numbers the opposite is true, and the fluctuations are overpowered by the viscous forces.
Turbulent flow is preferred for heat transfer, because the movement in the water cause rapid heat transfer.
2.4.3 Shear Stress
In order to evaluate different energy parts of a turbulent flow, shear stress,τw [Pa], that reduces the flow speed towards the wall of the pipe needs to be considered.
This is related to the velocity profile slope. Figure 2.13 shows this phenomenon.
Figure 2.13: Shear Stress
2.4.4 Darcy Friction Factor
Further, Darcy’s friction factor, f [−], can be found.
f = Wall friction force
Inertial force = 8τw
ρv2 [−] (2.13)
For laminar flows, the Darcy friction factor is only dependent on the Reynolds number, meaning thereof independent of the roughness of the pipe surface. Therefore equation 2.14 can be used.
f = 64
Re [−] (2.14)
This equation is true for when the water is flowing horizontally in a round pipe.
2.4.5 Colebrook Equation
For turbulent flows, evaluating the Darcy friction factor can be a bit more difficult.
Many scientist have tried to find a way to find the Darcy friction factor, and Cole-brook is one of them. He developed equation 2.15, for evaluating the friction factor
in a fully developed turbulent flow in a pipe, which is dependent of the Reynolds number and the relative roughness,ε/D [−].
√1 This equation is difficult to solve, but the matlab script colebrook.m and the excel-paper gives the same results.
Table 2.4: Roughness values for new commercial pipes Material Rougness
Table 2.4 shows the values for the roughness of the pipes, though these values have a huge uncertainty of 60 %.
Using the Moody chart, this is easier to evaluate.
2.4.6 Pressure Drop in Straight Pipes
Finally, the total pressure drop can be calculated by equation 2.16. This equation may be applied regardless of pipe cross section, roughness of surface, internal flows, and flow direction.
∆P =fL D
ρv2avg
2 [Pa] (2.16)
It is recommended that the total pressure drop in the pipe does not exceed 20 kPa for underfloor heating [70]. Higher pressure drops are acceptable for other applications. In one example for heat exchanger, a pressure drop of 50 kPa was acceptable [4]. 100 kPa is equal to 1 bar.
2.4.7 Pressure Drop in Bends
When using pipes to transfer fluids, bends are often needed to change the direction of the pipes and the fluid flow. Bends of different types cause pressure drops within the fluid, so for calculating the total pressure drop in pipes, bends need to be taken into consideration as well. [24]
Figure 2.14 shows the different factors that play a role when evaluating a pressure drop in a bend. Equation 2.17 shows how the values from the figure are put together in order to evaluate the pressure drop.
∆P =ζb
ρv2
2 [Pa] (2.17)
(a) Pressure Drop (b) Hexagon Figure 2.14: Pressure drop through bends and a hexagon
The drag coefficient, ζb [−], is taken from a graph. It is dependent on the ratio between the radius, r [m], of the bend, and the inner diameter, di [mm], of the pipe, found in figure 2.14a. The pressure drop is also dependent on the density and velocity of the fluid within the pipe.
Table 2.5: Drag coefficient for bends withr/di ratio of 5 for smooth pipes δ Drag coefficient
ζb [−]
45° 0.075
60° 0.085
90° 0.1
Figure 2.14b shows how a bend shown in figure 2.14a would fit for a pipe circuit formed as a hexagon. Every bend adds pressure loss. For a pipe running around a hexagon shaped path, theδ value is 30°, as shown. Table 2.5 shows theζb values for bends in pipes with ar/di ratio of 5.