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Developing flow correlations for different valve geometries using reference media for R-744.


In order to simplify and accelerate the research and production process of valves, new valve test procedures are discussed and investigated. One interesting option is the testing of the valve flow characteristic using an appropriate reference medium. The use of reference media is desirable for both the development of a prototype valve in a research laboratory and the valve inspection at the end of the assembly line in order to guarantee the demands on quality. The main objective is to find a medium that is easily available and manageable, non-toxic, and safe and that meets the required conditions with respect to valve flow similarity. Compressed air is the most basic fluid available in the majority of plants and therefore one interesting option for a reference fluid that can easily be used for the valve inspection at the end of the assembly line. One important advantage of the usage of air is that it can be completely removed after testing. Additional candidates for possible reference fluids are oils or hydrocarbon-based low-viscosity fluids such as ARAL 4005.

The investigated valve types presented in this paper are developed as control elements for carbon dioxide (R-744). R-744 is an important alternative refrigerant candidate for the next generation of mobile air-conditioning and heat-pump systems used in transcritical vapor compression cycles (Kim et al. 2004; EU 2006). Main application areas for control valves in mobile refrigeration systems using R-744 are the compressor control valve, the expansion valve for one- and two-evaporator systems, and 3/2-way switching valves for heat-pump systems (Muller 2007; Robin et al. 2003; Kim et al. 2004; Casson et al. 2003). This paper presents experimental data of three different prototype valves with cone-, ball-, and slide-valve geometries that were tested using R-744 and two reference media, ARAL 4005 and R-729, under one-phase flow conditions. Three different test rigs were used to carry out the tests using boundary conditions that ensure flow similarity regarding Reynolds number and Mach number. One aim is to find physical-based flow correlations to describe the flow behavior of the three investigated valve geometries with carbon dioxide (R-744), hydrocarbon-based low-viscosity fluid (ARAL 4005), and air (R-729) as fluid. The main goal is to find a correlation between the flow coefficients of both R-744 and ARAL 4005 and R-744 and R-729 to enable the transferring of flow coefficients measured for ARAL 4005 or R-729 into flow coefficients for R-744.

In the literature, several correlations for expansion valves with two-phase flow are available. Most of them were developed for orifice tubes or short orifices and thermostatic expansion valves used with common refrigerants such as R-22 (Kim et al. 2002) and R-134a (Sing et al. 2001; Bittle et al. 1998; Ding et al. 2000). Only a few correlations were developed for R-744 two-phase flow: Martin et al. (2006) derived a correlation from measurements with orifices and Lemke et al. (2005) presented a bypass orifice model used for cycle simulation. Pfafferott and Schmitz (2004) used a R-744 valve model based on the correlation presented in IEC 60534-2-1 (IEC 2000). There are only a few publications with a focus on compressor valves, such as those by Brandon (2000), Hunemorder (2005), Norris (2002), and Watanabe (2002).

Lou (2005) developed a model for a compressor control valve for dynamic cycle simulations but did not present flow correlations, whereas one-phase flow correlations for airflow through small valves for air bearings were presented in Donat (2006) and Dormann (2002). Idelchik (1986) and Wagner (2001) presented correlations for gas flow for different valve geometries. Boswirth (2002), Touber (1976), and Fagerli (1997) presented correlations to describe the dynamic valve flow through the compressor inlet and outlet valves. All correlations found in the literature cannot directly be used for the specific geometries of the valve prototypes investigated in this study without adapting coefficients.

Single-phase flow correlations for compressor control valves have been discussed by Xinjiang et al. (2002). They used nitrogen as the test medium and substitute for R-134a to validate the valve simulation model, but they did not conduct experiments with different media. Singh et al. (2001) used nitrogen as the test medium to determine manufacturing variability of orifices. A correlation between flow coefficients for one-phase flow of R-744 and other test media cannot be found in the available literature. This lack of detailed information about one-phase flow correlations and coefficients motivates the experiments carried out in this study.

In the first part of the next section, a review of the basic flow equations for compressible and incompressible one-phase flow from the literature is presented at a glance. Based on this approach, flow coefficients for the investigated geometries are experimentally identified for the different media, R-744, R-729, and ARAL 4005. To describe the main dependency of the flow coefficients, the coefficient behavior in pipes for laminar and turbulent flow is discussed in the following two parts of the next section. In the last part of the next section, the similarity of flow conditions is defined to evaluate the selected boundary conditions for the measurements.


Basic Flow Correlations for One-Phase Flow

For the calculation of one-phase valve flow through the compressor inlet and outlet valves, a semi-empirical model is often used. Xinjinang et al. (2002), Boswirth (2002), Touber (1976), and Fagerli (1997), for example, used the following Bernoulli approach for compressible flow involving friction:

m([DELTA]p) = [alpha]*[epsilon]*[A.sub.0]*[square root of (2*[rho]*[DELTA]p)] (1)

This approach is similar to the basic flow equation that is used for flow rate measurements with orifices according to DIN EN ISO 5167 (DIN 1995). Flow contraction and friction effects are taken into account by the flow coefficient [alpha], whereas effects due to the flow compressibility are considered by the expansion coefficient [epsilon]. The density, [rho], refers to the flow conditions before the valve, and [A.sub.0] refers to the smallest cross-sectional area in the valve. Typical values for [alpha] are presented in Figure 1. Using the effective flow area, [A.sub.eff] = [alpha] * [epsilon] * [A.sub.0], Equation 1 can be simplified to

m([DELTA]p) = [A.sub.eff]*[square root of (2*[rho]*[DELTA]p)]. (2)

Flow losses are also often evaluated using the drag coefficient,

[zeta] = [[[DELTA]p]/[[p/2][w.sub.0.sup.2]]], (3)

in lieu of the product of flow coefficient [alpha] and expansion coefficient [epsilon] (Spurk 1996; Zoebl and Kruschik 1982). The variable [w.sub.0] represents the velocity at the narrowest spot of the valve duct. The product of the flow coefficient [alpha] and the expansion coefficient [epsilon] can be written in the following form as a function of the drag coefficient [zeta]:

[alpha]*[epsilon] = [1/[square root of ([zeta])]] (4)

The pipe friction resistance, [lambda], is used in the literature to evaluate the flow losses in flows through pipes with length L and diameter D:

[lambda] = [D/L]*[zeta] (5)

The flow losses that result from the compressibility properties are taken into account by the expansion coefficient [epsilon]. A value of [epsilon] = 1 means incompressible flow and a value of [epsilon] < 1 means compressible flow. The following effects should be considered if compressible flow is studied (Boswirth 2002):

* flow expansion (reduced contraction compared to an incompressible flow)

* reduction of the density in the outgoing flow

In this study, the expansion coefficient [epsilon] is calculated according to ideal gas flow conditions. The mass flow rate can be calculated differing from Equation 1 based on the equation of Saint-Venant-Wantzel that takes the isentropic expansion of an ideal gas from a vessel into the environment into account (Bosnjakovic 1998; Rist 1996; Stephan and Mayinger 1992; Touber 1976):

m = [alpha]*[A.sub.0]*[square root of ([[2*[kappa]]/[[kappa] - 1]]*[p.sub.1]*[[rho].sub.1]([([p.sub.2]/[p.sub.1]).sup.[2/[kappa]]]) - [([p.sub.2]/[p.sub.1]).sup.[[[kappa] + 1]/[kappa]]])] (6)

Index 1 relates to the gas conditions in the vessel before the valve, whereas index 2 relates to conditions in the smallest cross section of the valve. The equation of Saint-Venant-Wantzel is often used to simulate the inlet and outlet flow in combustion engines (Pischinger 2002). The expansion coefficient can be calculated using Equations 1 and 6:

[epsilon] = [square root of ([[p.sub.1]/[[p.sub.1] - [p.sub.2]]]*[[kappa]/[[kappa] - 1]]*([([p.sub.2]/[p.sub.1]).sup.[2/[kappa]]] - [([p.sub.2]/[p.sub.1]).sup.[[[kappa] + 1]/[kappa]]])] (7)

The physical mechanism of the flow through pipes is well understood and frequently discussed in literature (Spurk 1996; Wagner 2001; Zoebl and Kruschik 1982). It is presented in the following section with the objective to aid in the similar understanding of the main aspects of the more complex flow behavior of valve flow.

Flow Equation for Laminar Pipe Flow

The pressure losses in a laminar flow through a pipe (Re < 2300) are exclusively determined by the fluid viscosity and can analytically be calculated based on the Newtonian shear stress function. The pressure losses are proportional to the velocity in the laminar case, as plotted in Figure 2. The roughness of the pipe wall has thereby no significant influence.


The flow coefficient in a laminar flow through a circular pipe can be calculated using the Hagen-Poiseuille equation:

[alpha]*[epsilon] = [square root of (Re)]*[1/8]*[square root of (L/2*R)] or [lambda] = [D/L]*[zeta] = 64*[1/Re] (8)

Figure 2 shows the trend of the pipe friction resistance versus the Reynolds number. In the left region, [lambda] decreases with 1/Re up to the critical Reynolds number. The trend in the right region of Figure 2 can be explained based on the theory of turbulent flow discussed in the next section.

The behavior of a typical flow is similar and is plotted in Figure 1: In the range of lower Reynolds numbers for laminar flow, [alpha] shows a significant dependency of the Reynolds number whereas nearly constant values for [alpha] can be observed at high Reynolds numbers.

Flow Equation for Turbulent Pipe Flow

The pressure loss in turbulent flow is about proportional to the square of the velocity. This behavior can be explained by the turbulent fluctuation of fluid particles vertical to the main flow direction. If a vertically flowing fluid particle with higher velocity comes into an area with lower velocity, the fluid particle loses its kinetic energy by a shock; the kinetic energy loss is proportional to the square of the velocity. The main contribution to the pressure loss in turbulent flow is caused by the Reynolds stress that shows a completely different behavior than the stress behavior that leads to friction in laminar flows.

Pipes used in technical applications are always more or less rough. While wall roughness hardly affects the pressure drop in laminar flow, its influence in the turbulent case is quite considerable if the mean protrusion height k is greater than the thickness of the viscous sublayer [delta]. The roughness can be fully characterized by k and [delta] or the roughness ratio k/[delta]. This definition leads to the following differentiation in three cases (Spurk 1996; Wagner 2001):

a. The contribution of the roughness to the pressure loss is negligible if the protrusion height k is so small that all protrusions are inside the viscous sublayer (k < [delta]) and the effect on the resistance is negligible. In this case, we speak of a pipe with a hydraulic smooth surface. The pipe friction resistance depends only on the Reynolds number [lambda] = f (Re):

[lambda] = 0.0054 + [0.3964/[Re.sup.0,3]] if 2*[10.sup.4] [less than or equal to]Re[less than or equal to]2*[10.sup.6](VDI 1997) (9)

a. The protrusion height is about as large as the thickness of the viscous sublayer k[congruent to][delta]. This range is called the transition area, and the pipe friction resistance depends on both the Reynolds number and k/[delta] so that [lambda] = f (Re, [delta]/k). Some protrusions are rising into the turbulent flow and produce an additional pressure drop compared with the hydraulic smooth pipe.

a. All protrusions are rising into the turbulent flow k > [delta]. In this case, we speak of a dynamically completely rough surface. The pipe friction resistance depends only on the ratio k/[delta] so that [lambda] = f ([delta]/k):

[lambda] = [1/[(2*lg(d/k) + 1.14).sup.2]] (Spurk 1996; Wagner 2001) (10)

Figure 2 shows the pipe friction resistance as a function of the Reynolds number for pipes with different roughness ratios calculated with the inner pipe diameter d and the roughness k. Pipes with rough surfaces show a constant pipe friction resistance [lambda] if the Reynolds number is higher than the value that is defined by the limit curve. Under such conditions, the pipe flow is turbulent. A similar behavior is reported in the literature for valves and can be seen in Figure 1, in which the flow coefficient [alpha] is plotted versus the Reynolds number. At high Reynolds numbers when the valve is passed through by turbulent flow, the flow coefficient becomes constant. A similar behavior can be observed for the flow coefficients calculated in this study.

Reynolds Number and Similarity of Flow Conditions

The measurements carried out within this study are evaluated based on the above-defined flow coefficients and the following definitions for the hydraulic diameter and the Reynolds and Mach numbers:

[d.sub.hyd] = [[4*[A.sub.0]]/[s.sub.0]] (11)

Re = [[[w.sub.0]*[d.sub.hyd]]/[upsilon]] (12)

Ma = [[w.sub.0]/a] (13)

[A.sub.0] represents the cross-sectional area at the narrowest spot of the valve and [s.sub.0] the corresponding circumference at the cross section. Each valve was investigated at three different valve lift positions leading to three different values for the hydraulic diameter [d.sub.hyd] and three different duty points in this system. The coefficient a stands for the medium dependent sound velocity.

In order to get similar flow conditions for the investigated media, each flow must have the same geometry and identical Reynolds numbers and Mach numbers:

[Re.sub.1][congruent to][Re.sub.2] and M[a.sub.1][congruent to]M[a.sub.2] (14)

The flow patterns of valve flows in identical valve geometries carried out with different fluids are expected to be similar if the flow conditions are similar in accordance with Equation 14. Even the measured flow coefficients and their dependency on the Reynolds number are expected to be similar. The boundary conditions presented in the next section are defined corresponding to the test rigs used and can be compared and evaluated applying similar flow conditions.


This section deals with the geometry of the investigated prototype valves and the description of the valve test facilities and discusses the specification of the boundary conditions.

Three different test prototype valves with cone-, ball-, and slide-valve geometries (Figure 3) were investigated using three different fluids: R-744, R-729, and ARAL 4005. Each prototype valve was tested in three different fixed valve lift positions under steady-state flow conditions.


Schematics of all test rigs for R-744, R-729, and ARAL 4005 are shown in Figures 4 and 5. The measurements with ARAL 4005 were carried out in an open cycle with a pump that generated the pressure at the valve inlet (Figure 4, top). Compressed air at ambient temperature was used for the measurements with R-729 (Figure 4, bottom). The inlet pressure at the test valve is adjusted by an additional control valve. The outlet pressure is defined by the ambient pressure. The mass flow rate at the inlet, the temperature at the inlet, and the particular outlet pressures are measured at both test rigs with appropriate transducers. The measurements with R-744 were carried out at a test rig within a closed cycle equipped with a compressor, a gas cooler, and an electrically heated evaporator unit. The prototype valve is located in a hot gas bypass between the compressor outlet and compressor inlet. The inlet and outlet pressures are controlled using two valves before and behind the prototype valve. Mass flow rate, differential pressure, outlet pressure, and inlet temperature are measured by a data acquisition system. The refrigerant contains a definite oil concentration of about 4%.



Specification of Boundary Conditions

In this study, three different valve geometries were investigated using two reference media, ARAL 4005 and R-729, with the objective to calculate flow coefficients for R-744 based on correlations between flow coefficients of the reference medium and R-744. The boundary conditions of the measurement points are specified by the operating conditions of the test rigs and are presented in Figure 6, which shows the Mach number versus the Reynolds number for the cone-valve geometry. Figure 6 compares R-744 with both reference media ARAL 4005 and R-729. The Mach numbers for all R-744 measurement points are smaller than 0.3. This allows a treatment based on incompressible flow theory, similar to the measurement points of ARAL 4005. The range of Reynolds numbers for the R-744 flow is much higher compared to the ranges for ARAL 4005 and R-729, but all measurement points are in the turbulent flow regime. The Mach number for the R-729 flow is higher than 0.3. Hence, compressible phenomena are not negligible. This is in contrast to R-744 and ARAL 4005, which can still be treated as incompressible fluids.


Error Analysis

To evaluate the measurement error, the deviation of [alpha] can be calculated by using Equation 1 in the following way:

[alpha] = [m/[[epsilon]*A*[square root of (2*[rho]*[DELTA]p)]]] = [alpha](m,A,[rho],[DELTA]p) (15)

[??][delta][alpha] [approximately equal to][absolute value of [[[partial derivative][alpha]]/[[partial derivative]m]]]*[delta]m + [absolute value of [[[partial derivative][alpha]]/[[partial derivative]A]]]*[delta]A + [absolute value of [[[delta][alpha]]/[[partial derivative][rho]]]]*[delta] [rho] + [absolute value of [[[delta][alpha]]/[[partial derivative][DELTA]p]]]*[delta] [DELTA]p (16)

Table 1 shows the measured variables with measurement principle, range, and error deviation for the R-744 test rig. The values for the error deviation can be applied for the ARAL 4005 and the air test rigs, as well. The overall deviation for the R-744 measurement using the cone-valve geometry leads to the following values depending on the value of the considered valve area: [delta][[alpha].sub.rel] = [+ or -]1 - [+ or -]6%.
Table 1. Measured Variables, Measuring Principles, Measurement Range,
and Deviation for the R-744 Test Rig

Measured         Measuring         Range            Error
Variable         Principle                        Deviation

Mass flow      Coriolis-force  0-0.022 kg/s     [+ or -]0.5%
rate m

Differential    Strain gauge      0-8 bar     [+ or -]0.008 bar

Flow area A     Displacement        0-4           [+ or -]3*
                                 [mm.sup.2]      [10.sup.-9]

Temperature T    Electrical    -40[degrees]C    [+ or -]0.3 K
                 resistance          to

Pressure p      Strain gauge     1-60 bar      [+ or -]0.1 bar


This section presents the experimental data generated with the valve test facilities specified in the previous section and the evaluated data based on the definitions and flow theory of earlier sections. Three different valve geometries are investigated using R-744 and two reference fluids to establish flow correlations.

If the expansion coefficient [epsilon] is extracted from Equation 7, the flow coefficient [alpha] can be calculated from Equation 1. The flow coefficient for the cone valve investigated with R-744 shows a constant trend in Figure 7. It shows a main dependency on the valve lift and has values between 0.7 and 0.9. It is obvious that the dependency on the Reynolds number is negligible in particular for higher Reynolds numbers. High Reynolds numbers in the range between 15,000 and 200,000 indicates definite turbulent flow conditions. The trend of the flow coefficient versus Reynolds number is similar to the flow behavior in completely rough pipes (Figure 2) and valves (Figure 1) such as was discussed in earlier sections and in accordance with values presented in the literature. In contrast to the measurement results presented in the literature for compressor inlet and outlet lamella valves plotted in Figure 1, [alpha] shows a maximum if it is plotted versus the valve area (Figure 10).



The value of the expansion coefficient [epsilon] is plotted in Figure 7 and varies between 0.9 and 1. This affirms the assumption of incompressible flow condition. The basic trend of the flow behavior leads to a Reynolds-number-independent [alpha] for all three valve geometries. The values of [alpha] depend on both the flow area and the particular valve geometry and are plotted in Figure 10 versus the flow area for each valve geometry.

The mass flow rate can be approximated based on the Bernoulli approach (Equation 1) applying a valve lift dependent constant flow coefficient according to Figure 8. The characteristic of the flow coefficient behavior measured for the ARAL 4005 flow is very similar to that of the R-744 flow and leads to constant values that are only valve-lift dependent. In contrast to both R-744 and ARAL 4005, the valve flow investigation with R-729 leads to different flow coefficient and expansion coefficient behavior.


The flow coefficient data for R-729 are plotted in Figure 9 and show an obvious dependency on both the Reynolds number and the valve lift. The dependency on the Reynolds number is linear.


Main reasons for the differing flow behavior are probably the compressible flow properties. The evaluation parameter of the compressibility is the expansion coefficient [epsilon] that is also plotted in Figure 9. The lower the expansion coefficient, the higher the influence on the compressible flow properties. The flow coefficient for the smallest valve lift increases from 0.5 to 1.2 as the expansion coefficient decreases from 0.7 to 0.2. The flow coefficient at the largest valve lift increases only from 0.7 to 0.8 due to the relative high value of the expansion coefficient that is in the range between 0.6 and 0.5. The dependency of the flow coefficient [alpha] on the Reynolds number decreases with increasing Reynolds number, whereas for a Reynolds number between 1000 and 6000, [alpha] depends significantly on the Reynolds number. This dependency is considerably declining at higher valve lifts that are linked to higher Reynolds numbers. It seems that for higher Reynolds numbers corresponding to larger valve opening areas, constant flow coefficients can also be achieved. This trend coincides with the literature about compressor inlet and outlet valves presented in Figure 2.

Figures 10 and 11 show the trend of the flow coefficient [alpha] versus the valve flow area for cone-, ball-, and slide-valve geometries for R-744 and ARAL 4005 as fluids. The curves for both fluids show similar trends for each valve geometry. The flow coefficient value of the slide valve is higher than that for low-flow areas due to a leakage along the control edge at low valve lifts. The flow coefficient values for the cone- and ball-valve geometries are significantly lower than 1. The flow coefficient trend for the cone valve shows a maximum for both fluids R-744 and ARAL 4005. In order to clarify the deeper physical dependency between the flow coefficient trend and the detailed flow pattern or the flow restriction, computational fluid dynamics simulations or sophisticated flow measurements are necessary.


A main objective of this investigation is to find a correlation between the flow coefficients of R-744 and one of the reference fluids. The aim is to enable the transfer of a flow coefficient measured with ARAL 4005 or R-729 into a flow coefficient for R-744. The previous discussion shows that the valve coefficient trend measured with R-744 is similar to the results measured with ARAL 4005 but completely different from the results gained with R-729. To find a correlation between the R-744 and ARAL 4005 fluids, the ratio of the average flow coefficients [[alpha].sub.R-744] and [[alpha].sub.ARAL],

[[pi].sub.[alpha]] = [[[alpha].sub.R-744]/[[alpha].sub.ARAL]] [approximately equal to] const, (17)

is defined and presented in Figure 12 for the cone-valve geometry. The flow coefficient ratio shows a constant value of about 1.1 without significant dependence of the valve area or valve lift. The average error deviation of [[pi].sub.[alpha]] is about [+ or -]6.5% and is presented in Figure 12 using error bars.


This simple correlation between the flow coefficients of both fluids can be interpreted as an indicator for the similarity of the flow properties even though the Reynolds numbers of both flow patterns differ by the power of ten.

It seems difficult to find a similar correlation between the flow coefficients of R-744 and R-729. In order to find a correlation, the similarity of flow conditions (Equation 14) can be used to calculate new boundary conditions. If R-729 or nitrogen is used at a higher pressure of about 70 bar at the valve inlet, similar Mach and Reynolds numbers can be achieved. These boundary conditions meet the conditions of flow similarity and might lead to promising results in finding a correlation between the flow coefficients of R-744 and R-729.


Three different prototype valves with cone-, ball-, and slide-valve geometries were tested using R-744 and two reference media, ARAL 4005 and R-729, under one-phase flow conditions. Three different test rigs were used to carry out the tests using boundary conditions that achieve more or less flow similarity regarding Reynolds number and Mach number. This investigation was motivated by the lack of literature data and correlations for the specific valve geometries tested with these reference media.

The evaluation of the flow coefficients for both fluids R-744 and ARAL 4005 show constant flow coefficients with a main dependency on the valve lift. The general trend of this curve--generated by the plot of flow coefficient versus Reynolds number--is similar for both fluids despite a difference in Reynolds number of about 10. But both fluid flows can be treated similarly as turbulent and incompressible flow. This general flow coefficient behavior is also presented in the literature dealing with incompressible turbulent flow through pipes with completely rough surfaces and compressor inlet and outlet valve flow. The mass flow rates for R-744 and ARAL 4005 can be approximated using the semi-empirical Bernoulli approximation. The ratio of the flow coefficient of R-744 and ARAL 4005 plotted versus the valve flow area leads to a constant value that represents a simple correlation. This correlation permits transfer of a flow coefficient measured with ARAL 4005 into a flow coefficient for R-744 and vice versa.

The investigated flow coefficients for R-729 show different behavior, and it seems to be difficult to find a similar correlation between R-744 and R-729 due to the compressibility of the R-729 flow. In order to find a correlation between R-744 and R-729, the similarity of flow conditions concerning Mach number and Reynolds number can be used to calculate new boundary conditions. If R-729 is used at a higher pressure of about 70 bar at the valve inlet, similar Mach and Reynolds numbers can achieved so that similar flow patterns and flow coefficient trends can be expected.

This investigation shows that ARAL 4005 is a usable reference medium for comparing valve flow investigations with R-744. The use of ARAL 4005 as reference media is possible for both the development of a prototype valve in a research laboratory and valve inspection at the end of the assembly line in order to guarantee the demanded quality. Compressed air in a pressure range typically available in the manufacturing environment can also be used for an end-of-line inspection, but it is not possible to derive a flow coefficient correlation between R-744 and R-729 that allows a precise flow coefficient conversion.


A = flow area, [m.sup.2]

a = sound velocity, m/s

D = diameter, m

[d.sub.hyd] = hydraulic diameter, m

k = protrusion height, m

L = length, m

Ma = Mach number

m = mass flow rate, kg/s

p = pressure, bar

[DELTA]p = differential pressure, bar

R = radius, m

Re = Reynolds number

s = circumference, m

T = temperature, K

w = flow velocity, m/s


[alpha] = flow coefficient

[delta] = thickness of viscous sublayer, m

[epsilon] = expansion coefficient

[kappa] = ratio of specific heats

[lambda] = pipe friction resistance

[nu] = kinematic viscosity, [m.sub.2]/s

[rho] = density, kg/[m.sup.3]

[zeta] = drag coefficient

[pi] = ratio between flow coefficients


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Norbert Stulgies

Jurgen Kohler, DrIng

Wilhelm Tegethoff, DrIng

Sven Forsterling, DrIng

Axel Muller, DrIng

Horst Kappler

Received April 12, 2007; accepted January 10, 2008

Norbert Stulgies is a doctoral student and Jurgen Kohler is a professor in the Institute of Thermodynamics, Department of Mechanical Engineering, Technical University of Braunschweig, Braunschweig, Germany. Wilhelm Tegethoff is the managing director and Sven Forsterling is a project engineer of TLK-Thermo GmbH, Braunschweig. Axel Muller and Horst Kappler are project engineers of Thomas Magnete GmbH, Herdorf, Germany.
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Author:Stulgies, Norbert; Kohler, Jurgen; Tegethoff, Wilhelm; Forsterling, Sven; Muller, Axel; Kappler, Hor
Publication:HVAC & R Research
Article Type:Report
Geographic Code:1USA
Date:May 1, 2008
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