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Capillary Viscometer and Vibrating Tube Densimeter for Simultaneous Measurements up to 70 MPa and 423 K Alfredo Pimentel-Rodas, Luis A. Galicia-Luna,* and José J. Castro-Arellano Laboratorio de Termodinámica, S.E.P.I.-E.S.I.Q.I.E., Instituto Politécnico Nacional, UPALM, Edif. Z, Secc. 6, 1ER piso, Lindavista C.P. 07738, México D. F., México ABSTRACT: New equipment has been designed and constructed to simultaneously measure dynamic viscosity and density on the basis of capillary flow technique and vibrating tube method, respectively, up to 70 MPa and 423.15 K. This apparatus has been built taking into account a number of physical parameters such as volumetric flow, length, and inner radius of the capillary tube and border effects. On the basis of the results of this work, densities for liquids are measured with relative combined expanded uncertainties of 0.07 % for water and ethanol, 0.7 % for hexane, 0.2 % for heptane, and 0.17 % for 1-pentanol and 1-heptanol. For the dynamic viscosity, the corresponding relative combined expanded uncertainties are 0.7 % for water and ethanol, 4.0 % for hexane, and 1.1 % for heptane, 1-pentanol, and 1-heptanol. In the calculated uncertainty for density and dynamic viscosity was included the contribution of the sample impurity. Experimental determinations of both dynamic viscosities and densities are performed, for several pure liquids, at the same conditions of temperature, pressure, and volumetric flow and recorded by means of an electronic device. Data reliability has been verified comparing the measured values with the available literature data for carbon dioxide, water, heptane, and ethanol up to 353 K and 30 MPa. The maximum deviations of the measured data compared to the literature data are ± 3 μPa·s for dynamic viscosity and ± 0.25 kg·m−3 for density. Furthermore, new experimental data are reported for hexane, 1-pentanol, and 1-heptanol up to 353 K and 30 MPa with a combined expanded uncertainty mentioned before. (2) pressure impulse flow types. Among these categories, those equipped with the following elements are mostly used ones in industry: oscillating body, vibrating body, falling body, and capillary viscometers. The measurement techniques mentioned are described below with emphasis on the associated uncertainty related to each equipment. It is not the aim to perform an exhaustive review of the viscosity measurement techniques. For details about the theory and experimental methodology the reader is referred to the original literature sources.7−22 Oscillating Body Viscometers. With these instruments it is possible to simultaneously determine density and dynamic viscosity of the fluid. This type of equipment has been extensively used in measurements of viscosity of both liquids and gases including high and low density gases, aqueous solutions, organic liquids, salts, and liquid metals. Reliable data can be obtained with an estimated relative standard uncertainty of 0.2 % at low pressures and of 0.5 % at pressures up to 30 MPa.7−11 Vibrating Body Viscometers. It is possible to use this measurement technique to simultaneously determine density and dynamic viscosity of fluids. It is worth noting that such

1. INTRODUCTION Accurate experimental data of phase equilibria and viscosities of polar and nonpolar fluids in wide ranges of temperatures and pressures are of great importance for a number of scientific developments and for technological applications, namely, for design of equipment and development of industrial processes as well as operation of the pumping, homogenization, evaporation, or dehydration units of industrial equipment. High pressure dynamic viscosity data are scarce in the open literature, and the corresponding measurements have been the subject of many efforts. It is, therefore, very important from the theoretical and experimental points of view to develop new experimental apparatuses and models for determining viscosities of fluids with low uncertainty.1,2 Looking into the literature,2−4 it can be found out that there is lack of accurate data with uncertainties within 1 % of dynamic viscosity of fluids at high pressures. Gupta2 and Nieto de Castro et al.3 have explained that the determination of viscosity of liquids within 1 % uncertainty is difficult and time-consuming. For these reasons, there is a need to develop a new apparatus able to provide reliable dynamic viscosity data of fluids. According to Nieto de Castro et al.,3 Viswanath et al.,5 and Shin and Keum,6 viscometers currently available could be grouped in two categories depending on the movement of the fluid with respect to its surrounding: (1) drag flow types and © 2015 American Chemical Society

Received: February 16, 2015 Accepted: November 4, 2015 Published: November 18, 2015 45

DOI: 10.1021/acs.jced.5b00152 J. Chem. Eng. Data 2016, 61, 45−55

Journal of Chemical & Engineering Data

Article

drop along the length of the tube and to the geometry.31 For a Newtonian fluid flowing across a section of cylindrical tube, the Hagen−Poiseuille equation states24,25,36

instruments have been used in measurements at extremely high pressures (close to 1 GPa) and at low temperatures (around 0.01 K) as well as of the viscosity of mixtures in the critical region and at room temperatures. The relative standard uncertainty with this equipment is in the range of 0.3−1.5 %.11−18 Rolling Body Viscometers. This type of viscometer is not as precise as the equipment just described. Nevertheless, they have been used by several authors for the determination of the viscosity of liquids and gases at temperatures up to 448 K and pressures up to 200 MPa. In general terms, these instruments show a relative standard uncertainty around 3 %.11,19−22 Capillary Flow Technique. Among the measurements techniques previously mentioned, rheological measurements of viscous flow through capillary viscometers are firmly established both in theory and in experimentation.10,11 The capillary-type viscometer has been perhaps the first viscometer. This instrument remains the most commonly used for the measurement of dynamic viscosity of Newtonian and nonNewtonian fluids6,23 due to their features such as high precision, simplicity, easy to adapt to severe conditions of operation, and, most importantly, an absolute instrument.24 The fundamental design is relatively simple and is based on the theory of laminar flow that passes through a capillary tube. The theory was developed by Poiseuille in 184025 and amended by Hagenbach26 in 1860. Literature is rich on the information about this class of instruments, the different modifications made to the equation of Poiseuille (for the flow of compressible fluids passing through the tube), and its effects on structural changes in different equipment.27−31 Full understanding of equations that describe the behavior of such systems is of vital importance for this study; that is why international sources have been thoroughly analyzed to supply and establish a sound theoretical basis.24,32−35 In this work, new equipment has been developed to determine the dynamic viscosity of pure liquids and liquid mixtures up to 70 MPa and 423 K. According to the modified Hagen− Poiseuille equation (eq 3) the dynamic viscosity calculation requires the density. For this reason, a densimeter was included into the measuring system with the objective of determining the density and dynamic viscosity simultaneously at the same conditions of temperature, pressure, and volumetric flow. This modification improves the measurement of the dynamic viscosity to achieve a relative combined expanded uncertainty of 0.7 % without the contribution of sample impurity. In order to test data reliability, dynamic viscosity and density of different fluids (carbon dioxide, water, heptane, ethanol) were measured and were compared to the published values in the literature. New results are provided for hexane, 1-pentanol, and 1-heptanol. Experimental data are reported at temperatures from (293 to 353) K and up to 30 MPa with an evaluated relative combined expanded uncertainty for dynamic viscosity and density of 0.7 % and 0.06 %, respectively, without the contribution of sample impurity. The effect of the sample impurity is detailed in the Equipment and Experimental Methodology section, and the relative combined expanded uncertainty for dynamic viscosity and density are reported in all tables for all liquids studied in this work.

ΔP =

8LQη πa 4

(1)

where η is the dynamic viscosity (Pa·s), a is radius of the tube (m), ΔP is the pressure drop across the tube (Pa), L is length of the tube (m), and Q is volumetric flow rate (m3·s). In this work, for practical purposes, the Q unit is cm3·min−1. The Hagen−Poiseuille equation is presented based on the following assumptions: (a) the flow is steady state, (b) the fluid is incompressible with constant properties, (c) the flow is laminar, (d) the fluid is Newtonian, (e) the fluid behaves as a continuum, (f) no external force field acts on the fluid, (g) the length-to-radius ratio of the tube is extremely large, (h) no slip occurs at the wall, and (i) the system is isothermal. In an actual system not all of the assumptions listed above can be satisfied completely. However, by proper design and operation of the viscometer, deviations from the ideal and simplified solution can be minimized and can be corrected by introducing correction factors. Correction by Kinetic Energy. The Hagen−Poiseuille equation is valid for a section of tube where the flow pattern of the fluid has been fully developed into a parabolic profile. The pressure difference (in the region of developed laminar flow) is measured between the inlet and the outlet of a capillary tube of finite length. Excessive pressure drop occurs in the entrance region where the fluid is accelerating to the fully developed profile, and irreversible work is caused by the sudden contraction of the fluid. This means that the effective pressure head used to overcome viscous forces is actually slightly smaller than the total pressure head. The loss of pressure head corresponding to the kinetic energy was first discussed by Hagenbach26 in the derivation of Poiseuille’s equation and is referred to as the Hagenbach correction. With this modification, the Hagen−Poiseuille equation becomes24,32,36 ηmod =

mρ Q πa 4ΔP − 8LQ 8πL

(2)

where m is Hagenbach factor, which is an experimental constant related to the variation of kinetic energy through the capillary tube and ρ is the fluid density. End-Effect Correction. The phenomenon which leads to the kinetic energy correction also results in the end effect or Couette correction.32 As the fluid proceeds from the bulb where its velocity is low to the capillary where its velocity is relatively high, there is a convergence of the stream, while at the exit end there occurs a divergence of the fluid. This makes an additional pressure drop,11,36 so that the capillary length must be replaced by an effective length (L + na), which depends on the capillary radius. The value of “n” is generally not much larger than unity, so that the end effect correction becomes negligible when L is much larger than “a”. With both the kinetic energy and the end-effect corrections, the Hagen−Poiseuille equation becomes

2. THEORY Design of the capillary tube viscometer is relatively simple based on the momentum and mass balance equations (particularly in the theory of laminar flow inside the cylindrical tube). The shear stress on the fluid is related to the pressure

ηcorr =

mρ Q πa 4ΔP − 8Q (L + na) 8π (L + na)

(3)

In this work, the value of the constant n is 0.69 ± 0.04 according to Kestin et al.35 for Reynolds numbers of 50 or less. 46



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Determination and Validation of the Radius of the Capillary Tube. Since the dynamic viscosity is calculated from the Hagen−Poiseuille equation depending on the fourth power of the radius of the tube (eq 3), it is very important to determine the used radius accurately. The mass of the capillary tube was determined at vacuum. Then, the tube under vacuum was loaded with water and a new weighing of the capillary tube was performed to know the increase in mass by water. The conditions of temperature and pressure were known; therefore, the density of water also was known. The internal radius of the capillary tube (stainless steel) was calculated considering the volume occupied by the mass of the water through the capillary tube. Mass measurements were conducted using an analytical balance by Sartorius LC1201S brand, which has a standard uncertainty of 1·10−7 kg. For mass determination at least seven weighings were carried out. Data of density of water at the measurement conditions of this work were taken from Wagner and Pruβ37 (relative standard uncertainty of 0.0001 %, at ambient pressure, and, in the most important part of the liquid region, from 0.001 % to 0.02 %). The calculated internal radius and its respective standard uncertainty is 1.259·10−4 m ± 7·10−7 m. Calibration of Pressure and Temperature Sensors. The system pressure was recorded using two pressure transducers (PDCR 4010-A093) connected to a digital display (DPI 145, Druck) with a resolution of ± 1·10−7 MPa. The pressure transducers were calibrated against a dead-weight balance (Desgranges & Huot, model 5304) with a certified precision of the order of ± 0.005 % (full scale). The transducer calibration covers two scales of pressure, the first with measurement range from 0 to 30 MPa, and the second with the range from 0 to 69 MPa. Regardless of the measurement range of the second pressure transducer, in this work measurements up to 30 MPa were carried out, so that the calibration of both transducers was until that value. For this work, after calibration of pressure transducers, a maximum deviation of the order of ± 0.0014 MPa was obtained. The expanded uncertainty (k = 2) in pressure was obtained corresponding to an estimated maximum of the order of 0.0027 MPa. The system temperature was measured by two platinum probes (Pt 100 Ω, Specitec) connected to a digital display (model F250, Automatic Systems Laboratories) with a resolution of ± 0.001 K. The temperature calibrations were made using a calibration system (Automatic Systems Laboratories F300) using a 25 Ω reference probe (model, Rosemount, 162CE ± 0.005 K). In this work, after calibration of platinum probes, a maximum deviation of ± 0.008 K was obtained, while the expanded uncertainties (k = 2) in temperature for both probes are 0.015 K. In both cases an Electronic Acquisition Data (EAD) was programmed with the aim to obtain the real-time data. The acquisition program was programmed in Agilent VEE PRO 7.0 software. This acquisition allows descriptive statistics, a very important tool to know the real stability of the system by means of the standard deviation of involved variables, as well to obtain representative data from the set of points to certain conditions. The EAD was used in all measurements. Calibration of Vibrating Tube Densimeter (VTD). The VTD (DMA HPM, mPDS 5 with CPU and transducer board, Anton Paar) was calibrated using the method described by Galicia-Luna et al.38 with nitrogen and water as fluids of

In our work, eq 3 is used to determine viscosities and the assumed value for “m” is 1.12.11,36 Furthermore, the effects of surface tension and viscous heating are considered negligible, and the discussion is limited to Reynolds number less than 50 at which simultaneous measurements of dynamic viscosity and density were carried out. As shown in eq 3, the dynamic viscosity determination depends on the density of the fluid. For this reason, a reliable and simple method to accurately determine density of the fluid under the same volumetric flow, pressure, and temperature was coupled with the experimental system. A vibrating tube densimeter (VTD) can satisfy all the required conditions mentioned so far.

3. EQUIPMENT AND EXPERIMENTAL METHODOLOGY Materials. Carbon dioxide research grade with a stated purity of 99.995 % and nitrogen research grade with a stated purity of 99.995 % were purchased from INFRA (México). Sigma-Aldrich supplied water with purity 99.9 % (HPLC grade), hexane with purity 96.7 % (anhydrous), heptane with purity of 99.3 % (anhydrous), 1- pentanol with purity 99.4 % (A. C. S. reagent), and 1-heptanol with purity 99.4 % (A. C. S. reagent). Ethanol with purity 99.9 % was acquired from MERCK with maximum content of water near 0.01 %. The samples’ purities were taken from the certificate of analysis provided by the manufacturers. Except for CO2 and N2, all compounds were carefully degassed by agitation under vacuum prior to injection into the system. The water contents for alkanes and alcohols were determined using a Karl Fischer coulometer (Metrohm, 831), and the results are hexane 367 ppm, heptane 552 ppm, ethanol 1232 ppm, 1-pentanol 2236 ppm, and 1-heptanol 1915 ppm. Standard uncertainty of water content is 14 ppm. Experimental Information. To develop this equipment, the following conditions were set: (i) viscosities up to 10 000 μPa·s, (ii) densities of less than 1050 kg·m−3, (iii) temperatures up to 423 K, and (iv) pressures up to 70 MPa for pure liquids and liquid mixtures. Some selected parameters such as length (L), radius (a) of the capillary tube, and the volumetric flow (Q) are defined based on specific criteria. The selection of the physical parameters of the capillary tube (length and radius) is important because it is related to achievement of developed laminar flow (L/2a > 300).24 For practical purposes it is convenient to select capillary tubes of considerable length combined with small radius and Reynolds numbers in laminar flow, thus reducing edge effects. Physical characteristics of capillary tube have been chosen according to the literature.24,36 Selection of the Radius of the Capillary Tube and Determination of the Length. The inner radius of the capillary was selected to be 1.27·10−4 m, as larger radius results in very small pressure drops, which cannot be determined with sufficiently low uncertainty. On the contrary, smaller internal radiuses are commercially difficult to obtain and require much more careful operation. In order to select the length of the tube (the relation of (L/2a > 300) should be satisfied here), some tests with capillary tubes of different length (0.45, 0.63, and 0.755) m were carried out to achieve the minimum uncertainty and the minimum difference between the measured data and those published in literature. The lengths of capillaries tubes were determined using a displacement transducer (model LS303C + ND 221, MicromaHeidenhain) with a standard uncertainty of 10 μm. 47



Article

Figure 1. Apparatus used in this work for the simultaneous determination of dynamic viscosity and density: (1) Temperature Indicator, (2) Pressure and Pressure Drop Indicator, (TRP1, TRP2) Pressure Transducers, (T1, T2) Platinum Probes, (SP) Syringe Pump, (VTD) Vibrating Tube Densimeter, (PC) Computer, (VP) Vacuum Pump, (V1, V2) Valves, (ESC) Purge.

reference. Reference density values of water and nitrogen were calculated using the equations of state developed by Wagner and Pruβ37 and Span et al.,39 respectively. Experimental densities were obtained according to the following equation:38 ρF (P , T ) = ρH O (P , T ) 2

+

[τF2(P , T ) − τH2O2(P , T )][ρH O (P , T ) − ρN (P , T )] 2

2

τH2O2(P , T ) − τN2 2(P , T )

(4)

where ρF(P, T) is the density of the fluid in study, ρH2O(P, T) and ρN2(P, T) are the densities of water and nitrogen, respectively, and τF(P, T) is the vibration period for each fluid in the study at the same conditions of pressure and temperature within experimental error. Experimental Uncertainty in the Determination of Density. The calculation of the uncertainty of the experimental densities was done as follows: The vibration periods (τ) for nitrogen, water, and the fluids under study (ethanol, hexane, heptane, 1-pentanol, and 1-heptanol) and calculated values of water, and nitrogen densities were obtained and recorded under the same pressure and temperature (the temperature is fixed and the vibration periods were determined at each pressure for all fluids investigated). In order to corroborate the experimental methodology, densities of heptane were measured experimentally. The maximum difference in temperature between the calibration of platinum probes and the temperature on measurements of density (for all compounds and each temperature of measurement) is ± 0.003 K. On the other hand, the maximum difference in pressure between the calibration of pressure transducers and the pressure on measurements of density (for all compounds and each measured pressure) is ± 0.002 MPa. The expression for the combined standard uncertainty in density measurements corresponds to the positive square root of the variance uc2 which is defined by eq 4. Since the density is determined at fixed temperature and pressure, within the experimental uncertainty, the combined standard uncertainty turns out to be only a function of water and nitrogen periods, water and nitrogen densities, and the studied

Figure 2. Relative deviations of the measured dynamic viscosities of carbon dioxide using different lengths and volumetric flows at 297 K from the correlation of Fenghour et al.49 (authors report a relative standard uncertainty of 2 % in dynamic viscosity in the measurement region); ■, L = 0.45 m, Q = 1 cm3·min−1; ●, L = 0.45 m, Q = 2.5 cm3·min−1; ▲, L = 0.45 m, Q = 5 cm3·min−1; ▼, L = 0.63 m, Q = 1 cm3·min−1; ◆, L = 0.63 m, Q = 2.5 cm3·min−1; □, L = 0.63 m, Q = 5 cm3·min−1; ○, L = 0.755 m, Q = 1 cm3·min−1; △, L = 0.755 m, Q = 2.5 cm3·min−1; ▽, L = 0.755 m, Q = 5 cm3·min−1.

fluid period. Therefore, using the NIST Technical Note 1297 (Appendix A, eq A-3),40 the combined standard uncertainty for the experimental density is expressed by ⎛ ∂ρ ⎞2 ⎛ ∂ρ ⎞2 F ⎟ 2 ⎜ ⎜ F⎟ 2 uc (ρ) = ⎜ ⎟ u (ρH2O ) + ⎜ ∂ρ ⎟ u (ρN2 ) ∂ ρ ⎝ H 2O ⎠ ⎝ N2 ⎠ 2

⎛ ∂ρ ⎞2 ⎛ ∂ρ ⎞2 2 F ⎟ 2 F ⎜ +⎜ ⎟ u (τH2O) + ⎜⎜ ∂τ ⎟⎟ u (τN2) ⎝ ∂τH2O ⎠ ⎝ N2 ⎠ ⎛ ∂ρF ⎞2 2 +⎜ ⎟ u (τF) ⎝ ∂τF ⎠ 48

(5) DOI: 10.1021/acs.jced.5b00152 J. Chem. Eng. Data 2016, 61, 45−55


Article

where each term is obtained by differentiation of eq 4, resulting in the following expressions: ⎛ ∂ρ ⎞2 ⎡ τ 2 − τ 2 ⎤2 N2 F ⎟ ⎜ ⎢ F ⎥ ⎜ ∂ρ ⎟ = ⎢ τ 2 − τ 2 ⎥ ⎣ H 2O N2 ⎦ ⎝ H 2O ⎠

Table 1. Experimental Values of Dynamic Viscosity and Density for Water Measured at Q = 1 cm3·min−1 with Capillary Tube Viscometer and Vibrating Tube Densimeter, Respectivelya T

(6)

⎛ ∂ρ ⎞2 ⎡ τ 2 − τ 2 ⎤2 H 2O ⎜ F ⎟ = ⎢− F ⎥ 2 2 ⎜ ∂ρ ⎟ ⎢ τ − τ ⎣ ⎦ H 2O N2 ⎥ ⎝ N2 ⎠

(7)

2 2 ⎛ ρ ⎛ ∂ρ ⎞2 − ρN ⎞ ⎡ τF 2 − τN 2 ⎤ H 2O 2 F 2 2 ⎢ ⎥ ⎜ ⎟ ⎜⎜ ⎟⎟ = ( −2τH O) ⎜ 2 2⎟ 2 2 2 ⎝ ∂τH2O ⎠ ⎝ τH2O − τN2 ⎠ ⎢⎣ τH2O − τN2 ⎥⎦

(8)

⎛ ρ ⎛ ∂ρ ⎞ − ρN ⎞ 2 ⎟ ⎜⎜ F ⎟⎟ = (2τN )2 ⎜⎜ H2O2 2⎟ 2 ∂ τ τ − τ ⎝ N2 ⎠ ⎝ H 2O N2 ⎠ 2

2

⎡τ2−τ 2⎤ H 2O ⎢ F ⎥ ⎢⎣ τH2O2 − τN2 2 ⎥⎦

2 ⎡ ρ − ρN ⎤ ⎛ ∂ρF ⎞2 2 ⎢ H 2O 2 ⎥ ⎟ = (2τF) ⎜ ⎢⎣ τH2O2 − τN2 2 ⎥⎦ ⎝ ∂τF ⎠

2

(9)

(10)

The relative combined expanded uncertainty in density for six compounds (water, hexane, heptane, ethanol, 1-pentanol, and 1-heptanol) is 0.06 % without the contribution of sample impurity.41 Considering the effect of the impurities of the samples in the experimental uncertainty,41−43 the corresponding relative combined expanded uncertainties for the density are 0.07 % for water and ethanol, 0.7 % for hexane, 0.2 % for heptane, and 0.17 % for 1-pentanol and 1-heptanol. Experimental Uncertainty in the Determination of Dynamic Viscosity. The uncertainty of experimental dynamic viscosity was calculated as follows: The density of fluids under study (water, ethanol, hexane, heptane, 1-pentanol, and 1-heptanol), pressure drop, and volumetric flow were recorded under the same conditions of pressures and temperatures within the experimental uncertainty (the conditions of temperature and pressure are similar to those mentioned in the last section). As the dynamic viscosity was measured at the same conditions of temperature, pressure, length of tube, inner radius, and volumetric flow, the combined standard uncertainty is only a function of pressure drop and fluid density. Therefore, as in the case of experimental density uncertainty,40 the combined standard uncertainty obtained for the experimental dynamic viscosity is

⎞2 ⎛ ∂ηF ⎞2 ⎛ πa 4 ⎜ ⎟ =⎜ ⎟ ⎝ ∂ΔP ⎠ ⎝ 8Q (L + na) ⎠

(13)

2.061 4.085 7.118 11.158 17.210 20.234 23.255 26.278 2.053 4.076 7.109 11.150 14.177 17.203 20.228 23.250 26.271 29.788 4.081 7.113 11.150 14.173 17.195 20.214 26.252 2.044 4.073 7.107 11.142 14.161 17.181 20.194 23.203 26.211 29.715

kg·m

−3

999.51 999.98 1001.28 1003.59 1006.18 1006.98 1008.25 1010.35 992.98 994.07 995.47 996.66 997.99 999.90 1000.99 1001.91 1003.18 1004.77 984.88 985.99 988.22 989.5 990.83 991.48 994.06 972.64 973.51 975.40 977.00 977.85 979.13 980.96 982.36 983.69 984.92

ηexp μPa·s 991.74 993.05 992.82 989.76 989.74 988.00 988.32 986.35 653.83 653.46 653.79 654.61 654.98 655.24 655.67 655.94 656.22 658.64 468.55 468.47 470.04 470.66 471.18 472.53 475.15 356.50 357.41 358.21 359.18 360.04 360.96 361.71 362.72 363.77 364.77

(14)

The relative combined expanded uncertainty (k = 2) for the dynamic viscosity determinations of six compounds (water, hexane, heptane, ethanol, 1-pentanol, and 1-heptanol) was evaluated to be 0.7 % without the contribution of sample impurity. Considering the effect of the impurities of the samples in the experimental uncertainty,42−47 the corresponding relative combined expanded uncertainties for the dynamic viscosity are 0.7 % for water and ethanol, 4.0 % for hexane, and 1.1 % for heptane, 1-pentanol, and 1-heptanol. Experimental Procedure and Apparatus. The density and dynamic viscosity were determined using different lengths and volumetric flows. The obtained data were compared with the data published in the literature trying to find the minimum

where each term is obtained by differentiation of eq 3, resulting in the following expressions:

(12)

MPa

⎛ ∂η ⎞2 ⎛ ⎞2 mQ ⎜⎜ F ⎟⎟ = ⎜ − ⎟ ⎝ 8π (L + na) ⎠ ⎝ ∂ρF ⎠

(11)

2 ⎛ ∂ηF ⎞2 ⎛ 3a 4nΔPπ 2 + 4a3LΔPπ 2 + mnQ 2ρ ⎞ ⎟ ⎜ ⎟ =⎜ ⎝ ∂a ⎠ 8Qπ (na + L) ⎠ ⎝

K 293.361 293.364 293.373 293.380 293.385 293.384 293.381 293.372 313.087 313.082 313.083 313.082 313.082 313.081 313.079 313.079 313.079 313.079 332.862 332.857 332.855 332.853 332.848 332.851 332.851 352.577 352.574 352.579 352.587 352.588 352.595 352.598 352.596 352.597 352.597

ρexp

a Standard uncertainties u are u(P) = 0.0014 MPa, u(T) = 0.008 K; relative combined expanded uncertainty for the density, Urc(ρexp) = 0.0007 and for dynamic viscosity, Urc(ηexp) = 0.007.

⎛ ∂η ⎞2 ⎛ ∂η ⎞2 uc 2(η) = ⎜ F ⎟ u 2(a) + ⎜ F ⎟ u 2(ΔP) ⎝ ∂a ⎠ ⎝ ∂ΔP ⎠ ⎛ ∂η ⎞2 + ⎜⎜ F ⎟⎟ u 2(ρF ) ⎝ ∂ρF ⎠

P

49



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Table 2. Experimental Values of Dynamic Viscosity and Density for Heptane Measured at Q = 1 cm3·min−1 with Capillary Tube Viscometer and Vibrating Tube Densimeter, Respectivelya T

Figure 3. Relative deviations of the measured dynamic viscosities of water at L = 0.755 m and Q = 1 cm3·min−1 from the correlation of reference;50 ■, T = 293.38 K; ●, T = 313.08 K; ▲, T = 332.85 K; ▼, T = 352.60 K.

Figure 4. Relative deviations of the measured densities of water from the correlation of Wagner and Pruβ;37 ■, T = 293.38 K; ●, T = 313.08 K; ▲, T = 332.85 K; ▼, T = 352.60 K.

deviation. For this, three volumetric flows and three lengths of capillary tube were chosen with the radius selected for the following four fluids, CO2, water, heptane, and ethanol using eq 3 with m = 1.12 and n = 0.69. The experimental procedure for the simultaneous determination of the dynamic viscosity and density of the fluids in liquid phase using the apparatus developed in this work is as follows: (a) The first step is to clean the experimental circuit. Then, under vacuum the system is filled with the fluid under study by the syringe pump. (b) The temperature of the experimental system is fixed until its stabilization within 0.005 K. (c) The system pressure is fixed. Once the system is stabilized, the fluid flows at a constant volumetric flow to achieve the desired conditions of temperature, pressure, and flow rate. Then, the system variables (temperature, pressure, volume flow, vibration period, and pressure drop) are recorded through a data acquisition unit. The same procedure is repeated until reaching the maximum value of pressure (for this work 30 MPa).

P

K

MPa

293.249 293.250 293.249 293.250 293.248 293.249 293.249 293.249 293.253 313.080 313.080 313.077 313.075 313.078 313.075 313.075 313.078 313.076 313.074 332.780 332.774 332.778 332.776 332.777 332.779 332.777 332.778 332.778 352.391 352.394 352.397 352.391 352.391 352.394 352.397 352.397 352.396 352.392

2.103 3.666 6.195 10.385 14.421 17.447 20.472 23.543 26.611 2.050 4.083 6.733 11.204 14.195 17.225 20.247 23.266 26.286 29.805 2.084 4.091 7.031 11.151 14.177 17.172 20.193 23.214 26.231 2.184 4.102 7.146 11.159 14.169 17.199 20.204 23.212 26.228 29.758

ρexp kg·m

−3

685.58 687.27 689.52 693.22 696.48 698.88 701.01 703.30 705.50 669.15 671.15 673.77 678.28 681.03 683.61 686.13 688.54 691.26 693.98 652.25 654.57 658.26 662.67 665.56 668.79 671.47 674.57 677.28 635.33 637.96 641.83 646.85 650.40 653.84 657.05 660.04 663.06 666.60

ηexp μPa·s 422.89 431.66 444.85 468.06 490.70 506.20 523.71 540.70 556.74 338.79 347.21 358.95 378.24 391.22 404.91 418.54 431.39 444.92 459.62 280.49 288.58 299.44 314.25 326.05 336.70 347.19 358.09 368.86 234.58 241.26 251.37 263.53 273.20 282.98 292.16 301.51 310.59 320.66

a

Standard uncertainties u are u(P) = 0.0014 MPa, u(T) = 0.008 K; relative combined expanded uncertainty for the density, Urc(ρexp) = 0.0020 and for dynamic viscosity, Urc(ηexp) = 0.011.

(d) Steps (b) and (c) are repeated at other temperatures. (e) The measured values are used in eqs 4 and 3 to calculate the density and dynamic viscosity, respectively. The main parts of experimental equipment are the syringe pump, the capillary tube, the vibrating tube densimeter (VTD), and the electronic acquisition data. A schematic flow diagram of the developed equipment is shown in Figure 1.

4. VALIDATION OF THE EXPERIMENTAL METHODOLOGY AND NEW RESULTS To validate the experimental methodology, various tests were carried out using the new equipment. The first tests were performed on carbon dioxide up to 297 K and 30 MPa, using lengths (0.45, 0.63, and 0.755) m combined each with 50



Article

Table 3. Experimental Values of Dynamic Viscosity and Density for Ethanol Measured at Q = 1 cm3·min−1 with Capillary Tube Viscometer and Vibrating Tube Densimeter, Respectivelya T

Figure 5. Relative deviations of the measured dynamic viscosities of ethanol at L = 0.755 m and Q = 1 cm3·min−1 from the data published by Assael and Polimatidou (authors report a relative standard uncertainty of 0.5 % for dynamic viscosity);16 ■, T = 298.15 K; ●, T = 323.15 K.

P

K

MPa

298.151 298.153 298.152 298.152 298.149 298.153 298.151 298.152 298.152 323.157 323.154 323.153 323.155 323.156 323.153 323.152 323.152 323.154 333.279 333.278 333.277 333.278 333.276 333.275 333.277 333.277 333.278

1.022 3.922 8.793 12.491 17.711 21.982 24.097 26.940 29.139 0.626 3.237 9.248 12.839 15.104 18.790 23.256 25.042 27.936 0.909 1.891 5.093 9.067 12.887 16.623 22.108 25.690 29.272

ρexp kg·m

−3

785.87 788.69 792.53 795.70 799.91 802.87 804.50 806.36 807.89 763.659 766.361 772.358 775.758 777.664 780.780 784.532 786.147 788.458 754.250 756.361 758.805 762.841 766.568 770.196 775.145 777.944 780.933

ηexp μPa·s 1086.1 1111.5 1147.9 1175.1 1213.3 1244.4 1259.5 1280.3 1296.3 697.83 711.04 743.53 761.57 773.24 792.81 815.32 823.56 836.74 587.22 591.64 606.53 625.10 642.11 659.24 684.05 698.45 713.33

a

Standard uncertainties u are u(P) = 0.0014 MPa, u(T) = 0.008 K; relative combined expanded uncertainty for the density, Urc(ρexp) = 0.0007 and for dynamic viscosity, Urc(ηexp) = 0.007. Figure 6. Relative deviations of the measured densities of ethanol from the data published by Assael and Polimatidou (authors report a relative standard uncertainty of 0.1 % for density);16 ■, T = 298.15 K; ●, T = 323.15 K.

In the case of water, there are no experimental data reported in the literature at the same conditions of temperature and pressure reported in this work. From the whole set of experiments performed at four temperatures (293.37, 313.08, 333.85, and 352.59) K and pressures up to 30 MPa (Table 1), only results with the lower relative combined expanded uncertainties in dynamic viscosity and density are reported at L = 0.755 m and Q = 1 cm3·min−1. Therefore, the measured data were compared to calculated values of the dynamic viscosity and density resulting from the equations of Huber et al.50 and Wagner and Pruβ,37 respectively. For both equations, the authors claim a relative expanded uncertainty of 1 % for dynamic viscosity and a relative standard uncertainty between 0.001 % and 0.003 % for density. Furthermore, Figures 3 and 4 indicate that deviations of the measured values with respect to those published in literature are ± 0.4 % and ± 0.04 % for dynamic viscosity and density, respectively. Water experimental data confirm and validate the length of the tube and the volumetric flow proposed for the above case of carbon dioxide. At the same conditions (Q, L) as in the case of water, the dynamic viscosity and the density of heptane were determined. Values for temperature and pressure are given additionally. The results for L = 0.755 m and Q = 1 cm3·min−1, shown in Table 2, were compared with data published in the literature51,52 (the authors reported a relative standard uncertainty in

volumetric flows of (1, 2.5, and 5) cm3·min−1 and densities calculated by Span and Wagner.48 According to the results of these measurements, the lowest combined expanded uncertainty was calculated with the largest length (0.755 m) and the lowest volumetric flow (1 cm3·min−1). A comparison between the experimental data obtained in this work and the data published in literature49 (the authors have reported a relative standard uncertainty of 2 % in dynamic viscosity in the measurement region) is shown in Figure 2. Since the modified Hagen−Poiseuille equation requires density values, a vibrating tube densimeter was coupled to the equipment with the aim to obtain the density at the same conditions and consequently to improve the determination of the dynamic viscosity. Dynamic viscosity and density of water, ethanol, and heptane with L = (0.45, 0.63 and 0.755) m and Q = (1, 2.5, and 5) cm3·min−1 at four temperatures (293, 313, 333, and 353) K and pressures up to 30 MPa were measured to confirm and to validate the experimental methodology and experimental uncertainties of the measurements. 51



Article

respect to those reported in the literature at two temperatures at 298 and 323 K. The deviations from the experimental data reported in the literature16,53 (the authors report a relative standard uncertainty of 0.5 % for dynamic viscosity and 0.1 % for density) are shown in Figures 5 and 6 for dynamic viscosity and density, respectively. The maximum deviations are ± 0.23 % in dynamic viscosity and ± 0.02 % in density. The dynamic viscosity and density reported were determined at pressures up to 30 MPa (Table 3). Finally, new data of dynamic viscosity and density of hexane, 1-pentanol, and 1-heptanol were determined at temperatures between 293 to 353 K and pressures up to 30 MPa. Viscosity and density of hexane were determined at four temperatures (293.22, 313.07, 333.00, and 353.15) K and pressures up to 30 MPa (Table 4). In this case, the minimal deviations are obtained comparing our data with the calculated values in literature51,54 (the authors report relative uncertainties of 2 % and 0.04 % for dynamic viscosity and for density, respectively). Deviations in dynamic viscosity (maximum deviation of ± 0.52 %) and density (maximum deviation ± 0.04 %) are shown in Figures 7 and 8, respectively.

Table 4. Experimental Values of Dynamic Viscosity and Density for Hexane Measured at Q = 1 cm3·min−1 with Capillary Tube Viscometer and Vibrating Tube Densimeter, Respectivelya T

P

K

MPa

293.224 293.224 293.222 293.224 293.225 293.225 293.225 293.226 293.224 293.226 293.223 293.226 313.101 313.100 313.101 313.101 313.101 313.101 313.101 313.100 313.100 313.100 313.102 313.103 333.001 333.002 333.002 333.002 333.001 333.002 333.001 333.000 333.004 333.000 333.001 333.003 353.147 353.145 353.146 353.144 353.147 353.146 353.144 353.146 353.145

1.999 3.994 5.999 8.007 10.003 13.000 16.006 19.000 22.004 25.002 28.004 30.001 2.005 3.997 5.999 8.000 9.999 12.997 15.999 18.003 21.999 24.998 28.001 30.004 2.004 4.002 6.003 8.005 10.008 12.998 16.002 19.000 21.993 25.001 28.000 30.005 2.003 4.001 5.997 8.003 10.009 13.000 15.997 19.000 21.998

ρexp kg·m

−3

661.40 663.53 665.43 667.57 669.41 672.26 675.20 677.83 680.38 682.87 685.01 686.72 643.60 645.97 648.24 650.44 652.57 655.73 659.00 661.04 664.77 667.58 670.32 672.03 624.85 627.59 630.29 632.82 635.78 639.36 642.74 646.04 649.32 652.41 655.31 657.29 605.15 608.43 611.57 614.97 618.00 622.08 625.92 629.87 633.46

ηexp μPa·s 319.61 328.66 339.04 348.32 357.63 371.33 385.07 399.03 413.52 428.04 442.75 452.10 261.44 270.16 277.42 285.32 293.41 305.01 316.50 324.33 339.89 350.43 363.05 370.03 219.19 225.99 232.78 239.58 246.70 256.48 266.69 276.63 286.35 296.01 305.80 312.80 184.85 191.09 197.01 203.03 209.12 217.87 226.44 235.22 243.80

Figure 7. Relative deviations of the measured dynamic viscosities of hexane at L = 0.755 m and Q = 1 cm3·min−1 from the correlation of ref 51: ■, T = 293.22 K; ○, T = 313.10 K; ●, T = 333.00 K; □, T = 353.15 K.

a

Standard uncertainties u are u(P) = 0.0014 MPa, u(T) = 0.008 K; relative combined expanded uncertainty for the density, Urc(ρexp) = 0.007 and for dynamic viscosity, Urc(ηexp) = 0.04.

dynamic viscosity of 2 % and up to 0.5 % in density), and a maximum deviation of ± 0.4 % and ± 0.03 % for dynamic viscosity and density, respectively, was obtained. These data confirm the selection of values of the length and volumetric flow. Ethanol is the last fluid studied at the same conditions of water (length of the tube and volumetric flow). For this purpose, the data obtained in this work are compared with

Figure 8. Relative deviations of the measured densities of hexane from the correlation of ref 51: ■, T = 293.22 K; ●, T = 313.10 K; ▲, T = 333.00 K; ▼, T = 353.15 K; from experimental data:54 □,, T = 323.05 K; ○ T = 332.90 K. 52



Article

New experimental data of viscosities of 1-pentanol and 1-heptanol were determined at two temperatures (298.15 and 323.15) K and pressures up to 30 MPa, and the results are reported in Tables 5 and 6, respectively. The study of the behavior of liquid dynamic viscosity and density as a function of temperature (up to 353 K) and pressure (up to 30 MPa) validates and confirms the experimental methodology and equipment developed at the mentioned conditions. Experimental determinations for liquid mixtures are carried out at the same conditions and will be published in the near future.

Table 5. Experimental Values of Dynamic Viscosity and Density for 1-Pentanol Measured at Q = 1 cm3·min−1 with Capillary Tube Viscometer and Vibrating Tube Densimeter, Respectivelya T

P

K

MPa

298.151 298.152 298.153 298.150 298.151 298.150 298.149 298.150 298.151 298.151 323.150 323.152 323.149 323.153 323.154 323.152 323.151 323.152 323.151 323.153

2.003 3.999 6.003 8.005 10.001 13.000 16.004 19.006 22.000 24.998 1.999 4.001 6.001 8.000 9.998 12.999 16.001 19.002 22.001 25.002

ρexp kg·m

−3

811.93 813.30 814.64 815.96 817.26 819.15 821.00 822.80 824.56 826.27 794.19 795.77 797.32 798.83 800.31 802.47 804.56 806.59 808.56 810.48

ηexp μPa·s 3592.5 3661.2 3731.1 3798.6 3866.8 3971.8 4080.4 4189.9 4300.9 4410.0 1834.8 1868.3 1903.2 1937.4 1972.1 2024.9 2078.1 2133.9 2190.0 2245.0

6. CONCLUSIONS A new apparatus for simultaneous experimental determination of dynamic viscosity and density of liquids was developed. The equipment is able to determine dynamic viscosity and density, in liquid phase, up to 353 K and 30 MPa with a relative combined expanded uncertainty of 0.7 % in dynamic viscosity and density of 0.06 % without the contribution of sample impurity. Considering the effect of the impurities of the samples in the experimental uncertainty,41−47 the corresponding relative combined expanded uncertainties for density are 0.07 % for water and ethanol, 0.7 % for hexane, 0.2 % for heptane, and 0.17 % for 1-pentanol and 1-heptanol. For the dynamic viscosity, the corresponding relative combined expanded uncertainties are 0.7 % for water and ethanol, 4.0 % for hexane, and 1.1 % for heptane, 1-pentanol, and 1-heptanol. The dynamic viscosity and density data for water, heptane, and ethanol were used to validate the experimental methodology and to test three lengths of tube and three volumetric flows. The results for the viscosity were obtained with L = 0.755 m, Q = 1 cm3·min−1, a = 1.259·10−4 m, m = 1.12, and n = 0.69. The maximum deviations of the measured values with respect to those published in literature are ± 0.52 % and ± 0.04 % for dynamic viscosity and density, respectively. New experimental data for dynamic viscosity and density were presented for hexane, 1-pentanol, and 1-heptanol at temperatures between (293 to 353) K and pressures up to 30 MPa with the same relative combined expanded uncertainty evaluated. Finally, it is worth mentioning that the purity of the sample plays an important role in the experimental uncertainty reported.


Table 6. Experimental Values of Dynamic Viscosity and Density for 1-Heptanol Measured at Q = 1 cm3·min−1 with Capillary Tube Viscometer and Vibrating Tube Densimeter, Respectivelya T

P

K

MPa

298.151 298.153 298.152 298.151 298.151 298.150 298.153 298.152 298.152 298.149 323.149 323.153 323.152 323.151 323.150 323.152 323.151 323.151 323.151 323.150

2.002 3.999 6.006 8.000 10.002 12.997 15.999 18.995 22.001 25.000 2.006 4.003 6.000 7.993 10.000 13.006 16.007 19.004 22.008 25.007

ρexp kg·m

−3

819.81 821.20 822.30 823.70 824.80 826.41 828.31 831.30 833.20 835.10 802.09 803.50 804.98 806.39 807.79 809.85 811.88 813.79 815.68 817.57

ηexp μPa·s 5986.2 6110.2 6233.2 6359.2 6486.9 6682.8 6883.0 7086.0 7297.3 7511.5 2778.5 2832.4 2886.9 2942.1 3000.0 3085.0 3172.8 3262.5 3354.0 3446.0

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (Luis A. Galicia-Luna). Funding

The authors would like to thank the Instituto Politécnico Nacional and CONACyT for the financial support of this research. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Special thanks to Dr. Ali Eslamimanesh for his fruitful discussions.


53

NOTATION P = pressure T = temperature ΔP = pressure drop L = length of the capillary tube DOI: 10.1021/acs.jced.5b00152 J. Chem. Eng. Data 2016, 61, 45−55


Article

Q = volumetric flow η = dynamic viscosity ρ = density a = radius of capillary tube m = Hagenbach correction, constant n = end effect correction, constant τ = vibration period 2 uc = variance Urc = relative combined expanded uncertainty

(18) Caudwell, D. R.; Trusler, J. P. M.; Vesovic, V.; Wakeham, W. A. The Viscosity and Density of n-Dodecane and n-Octadecane at Pressures up to 200 MPa and Temperatures up to 473 K. Int. J. Thermophys. 2004, 25, 1339−1351. (19) Zéberg-Mikkelsen, C. K.; Baylaucq, A.; Watson, G.; Boned, C. High-Pressure Viscosity Measurements for the Ethanol + Toluene Binary System. Int. J. Thermophys. 2005, 26, 1289−1302. (20) Zéberg-Mikkelsen, C. K.; Watson, G.; Baylaucq, A.; Galliéro, G.; Boned, C. Comparative Experimental and Modeling Studies of the Viscosity Behavior of Ethanol + C7 Hydrocarbon Mixtures versus Pressure and Temperature. Fluid Phase Equilib. 2006, 245, 6−19. (21) Schaschke, C. J.; Abid, S.; Fletcher, I.; Heslop, M. J. Evaluation of a Falling Sinker-Type Viscometer at High Pressure Using Edible Oil. J. Food Eng. 2008, 87, 51−58. (22) Sato, Y.; Yoshioka, H.; Aikawa, S.; Smith, R. L., Jr. A Digital Variable-Angle Rolling-Ball Viscometer for Measurement of Viscosity, density, and Bubble-Point Pressure of CO2 and Organic Liquid Mixtures. Int. J. Thermophys. 2010, 31, 1896−1903. (23) Sisko, A. W. Capillary Viscometer for Non-Newtonian Liquids. J. Colloid Sci. 1960, 15, 89−96. (24) Kao, J. T. F.; Ruska, W.; Kobayashi, R. Theory and Design of an Absolute Viscometer for Low Temperature-High Pressure Applications. Rev. Sci. Instrum. 1968, 39, 824−834. (25) Poiseuille, J. L. M. Recherches Expérimentales Sur Le Mouvement Des Liquides Dans Les Tubes De Très Petits Diamètres; II. Influence De La Longueur Sur La Quantité De Liquide Qui Traverse Les Tubes De Très Petits Diamètres; III. Influence Du Diamètre Sur La Quantité De Liquide Qui Traverse Les Tubes De Très Petits Diamètres. C. R. Acad. Sci. 1840, 11, 1041−1048. (26) Hagenbach, E. Uber die Bestimmung der Zähigkeit einer Flüssigkeit durch den Ausfluss aus Röhren. Ann. Phys. 1860, 185, 385− 426. (27) Tohidi, B.; Todd, A. C.; Danesh, A.; Burgass, R. W.; Gozalpour, F. Viscosity and Density of Methane + cis-Decalin from 323 to 423 K at pressures to 140 MPa. Int. J. Thermophys. 2001, 22, 1661−1668. (28) Cunningham, D. B.; Doe, P. H.; Joshi, S. D.; Moradi-Araghi, A. Capillary Viscometer for Evaluating Low-Viscosity Solutions at Elevated Temperatures. Rev. Sci. Instrum. 1986, 57, 2310−2314. (29) Ripple, D. A Compact, High-Pressure Capillary Viscometer. Rev. Sci. Instrum. 1992, 63, 3153−3155. (30) Yamasaki, T.; Irvine, T. F. A Comparative Capillary Tube Viscometer to Measure the Viscous Properties of Newtonian and Power-Law Fluids. Exp. Therm. Fluid Sci. 1990, 3, 458−462. (31) White, J. P.; Davidson, V. J.; Otten, L. A Capillary Viscometer for Characterization of Fluid Foods. Food Res. Int. 1993, 26, 109−113. (32) Kaplan, N. P.; Colowick, N. P.; Hirs, C. H. W.; Timasheff, S. N. Methods in Enzymology; Academic Press: New York and London, 1973. (33) Galvin, G. D.; Hutton, J. F.; Jones, B. Development of a HighPressure, High-Shear-Rate Capillary Viscometer. J. Non-Newtonian Fluid Mech. 1981, 8, 11−28. (34) Vrentas, J. S.; Vrentas, C. M. Mechanical Energy Balances for a Capillary Viscometer. J. Non-Newtonian Fluid Mech. 1983, 12, 211− 224. (35) Kestin, J.; Sokolov, M.; Wakeham, W. A. Theory of Capillary Viscometers. Appl. Sci. Res. 1973, 27, 241−264. (36) Bingham, E. C. Fluidity and Plasticity; McGraw Hill Book Company: New York, 1922. (37) Wagner, W.; Pruβ, A. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. J. Phys. Chem. Ref. Data 2002, 31, 387−535. (38) Galicia-Luna, L. A.; Richon, D.; Renon, H. New Loading Technique for a Vibrating Tube Densimeter and Measurements of Liquid Densities up to 39.5 MPa for Binary and Ternary Mixtures of the Carbon Dioxide-Methanol-Propane System. J. Chem. Eng. Data 1994, 39, 424−431. (39) Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W. A Reference Quality Equation of State for Nitrogen. Int. J. Thermophys. 1998, 19, 1121−1132.

Subscripts

mod = modified corr = corrected F = fluid under study H2O = water N2 = nitrogen ref = reference vac = vacuum exp = experimental variable

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