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This power dissipation mostly goes into heat generation within the fluid. In most cases, the portion of heat exchange with the external environment (through the solid walls of the components in the system, including the orifice) is minimal and negligible. The temperature increase of the fluid can be calculated from the energy balance:
Figure 4.5 Hydraulic symbol of two valves and equivalent orifice networks.
The Eq. (4.9) points out how the effects due to the thermal expansion of the fluid can be neglected. This is a reasonable assumption for oils, which have a thermal expansion coefficient, α, typicall of 7 · 10−4 [1/K].
Assuming common properties for mineral based oil, the value of ΔT is approximately 5 to 6 degrees Celsius per 100 bar of pressure drop.
Example 4.1 Orifice Flow, Power Dissipation and Temperature Rise
An orifice is used in a pilot line of a system, connected to tank. The pressure in the line, upstream the orifice, is 190 bar, while the tank pressure is atmospheric. If the diameter of the orifice is D = 0.5 mm, evaluate the flow rate lost through it at maximum pressure and the power dissipated through it. Assume oil density ρ = 850 kg/m3 and Cf = 0.62. If the constant specific heat of the oil is 1.8 kJ/kg K, estimate the temperature rise for the fluid across the orifice.
Given:
The pressure drop across the orifice ΔpOR = 190 bar; the orifice diameter D = 0.5 mm; the orifice coefficient Cf = 0.62; the fluid density ρ = 850 kg/m3 and the specific heat coefficient cp = 1.8 kJ/kg K
Find:
1 the flow rate through the orifice QOR
2 the power dissipated by the orifice POR
3 temperature rise of the fluid through the orifice
Solution:
The ISO schematic of the system can be represented by the figure below. Note that the pilot line is represented as dashed line, according to the standard of representation.
1 QOR can be simply calculated by using the orifice equation (4.5), being the Δp across the orifice given by the problem data
2 The power dissipated is calculated from Eq. (4.8):
3 The temperature rise experienced by the fluid through the orifice can be estimated from Eq. (4.9), under the assumption that the entire heat dissipation increases the internal energy of the fluid
4.4 Parallel and Series Connections of Orifices
In some hydraulic circuits, orifices appear in series or in parallel configuration (Figure 4.6). In these cases, it can be useful to evaluate the overall behavior of the system of orifices.
One way to address this problem is to consider the definition of hydraulic resistance for an orifice. Then, consider that
and
where the hydraulic resistance of the orifice has the expression of Eq. (4.7).
From Eqs. (4.10) and (4.11), the area of the equivalent orifice can be calculated. Another way for determining the area of the equivalent orifice is to directly derive the relationship between Q and Δp of the set of orifices, as shown in the figure.
Figure 4.6 Orifice in parallel and in series.
For orifices connected in parallel, the overall flow is the sum of the individual flows, while the upstream and downstream pressures are equal for all:
(4.12)
Therefore, if the same flow coefficient is assumed for all orifices, the area of the equivalent orifice is represented by the sum of the individual areas, and the equivalent diameter is square root of the sum of the square of the individual diameters:
(4.13)
