Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 5, 2015

Primary standard for liquid flow rates between 30 and 1500 nl/min based on volume expansion

Peter Lucas, Martin Ahrens, Jan Geršl, Wouter Sparreboom and Joost Lötters


An increasing number of microfluidic systems operate at flow rates below 1 μl/min. Applications include (implanted) micropumps for drug delivery, liquid chromatography, and microreactors. For the applications where the absolute accuracy is important, a proper calibration is required. However, with standard calibration facilities, flow rate calibrations below ~1 μl/min are not feasible because of a too large calibration uncertainty. In the current research, a traceable flow rate using a certain temperature increase rate is proposed. When the fluid properties, starting mass, and temperature increase rate are known, this principle yields a direct link to SI units, which makes it a primary standard. In this article, it will be shown that this principle enables flow rate uncertainties in the order of 2–3% for flow rates from 30 to 1500 nl/min.

Corresponding author: Peter Lucas, VSL, Dutch Metrology Institute, Thijsseweg 11, 2629 JA, Delft, The Netherlands, E-mail:


The research leading to the results discussed in this report has received funding from the European Metrology Research Program (EMRP) (Grant/Award Number: HLT07) The EMRP is jointly funded by the EMRP participating countries within Euramet and the European Union.

Appendix A

We denote m(t) as the mass of water contained in the reservoir and the submerged part of the capillaries. This mass is a function of time and the density ρMUT of the water just upstream of the MUT. The volumetric flow rate through the MUT is then given by

(3)Q=-1ρMUTdmdt. (3)

The mass can be expressed as the integral of the water density ρ(x, t) over the volume V(t) (water inside the system). The volume depends on time since the system boundaries change due to the change in temperature. The time derivative of the water mass inside the system is then given as

(4)dmdt=ddtV(t)ρ(x,t)dV=V(t)ρ(x,t)tdV+V(t)ρ(x,t)ξ(x,t)dS, (4)

where ∂V(t) is the boundary surface of the volume V(t) and ξ(x,t) is the velocity of the points at the boundary of the system. For example, in the titanium reservoir, ∂V(t) is the boundary surface between water and titanium and ξ(x,t) is the velocity of the points at this boundary, which are moving due to the thermal expansion of the titanium reservoir. We can split the integration volume V(t) into two parts: one part is denoted VH(t) and has a positive temperature increase rate (heated fluid elements); the second one is denoted VC(t), which has a negative temperature increase rate (cooling down fluid elements once left the temperature-controlled bath). In the same way, the integral over the boundary ∂V(t) can be split into the sum of integrals over ∂VH(t) and ∂VC(t). Therefore, we have

(5)dmdt=VH(t)ρ(x,t)tdV+VH(t)ρ(x,t)ξ(x,t)dS+VC(t)ρ(x,t)tdV+VC(t)ρ(x,t)ξ(x,t)dS. (5)

In this equation, the first term corresponds to the volume expansion of water in the heated part of the system; the second term is a correction for the volume expansion of the system itself (negative flow rate) and the last two terms are corrections for the cooled part of the system. In the section CFD Simulations, it is argued that these cooling part corrections can be neglected and therefore we do not consider them further.

The thermal conductivity of titanium and steel is large as compared with water. Therefore, the temperature at ∂VH(t) inside the system will be very close to the temperature outside the system. This also implies that the water density can be considered constant at this surface at a given time. We denote this density ρVH(t). Hence, the second term of Eq. 5 can be written as

(6)VH(t)ρ(x,t)ξ(x,t)dS=ρVH(t)VH(t)ξ(x,t)dS=ρVH(t).dVH(t)dt. (6)

The volume of the heated part of the system VH(t) is given by its measured reference value for certain temperature T0 and by the thermal expansion of the materials of the heated part (titanium for reservoir and stainless steel for the connecting capillary). Owing to the fast heat transfer in the metal parts, the temperature does not change in the metal elements at a given time. Hence, we denote TR(t) as the temperature of the metal parts and we consider it constant in space. The density of the water at the boundary between the metal parts and the water can therefore be expressed as ρVH(t)=ρ(TR). Next, we denote βT and βS as the volume thermal expansion coefficients of the titanium and the stainless steel both at the temperature T0. We obtain

(7)VH(t)=VT(T0)(1+βT(TR(t)-T0))+VS(T0)(1+βS(TR(t)-T0)), (7)

where VT(T0) and VS(T0) are the volumes of water-filled cavities inside the titanium reservoir and inside the stainless steel capillary at temperature T0. These volumes will be known from differential mass measurements. TR(t) is measured by thermometer installed in a copper mounting attached to the titanium part. Eq. 7 can be rewritten as

(8)VH(t)=VH0(1+β(TR(t)-T0)), (8)

where VH0=VT(T0)+VS(T0) is the total reference volume at T0 and β is given by

(9)β=VT(T0)βT+VS(T0)βSVT(T0)+VS(T0)=βT+VS(T0)VT(T0)+VS(T0)(βS-βT). (9)

The difference βS-βT is of the same order of magnitude as βT, and the capillary volume contributes to the total volume by not more than 0.1% for the reservoir considered here. The contribution of the second term of Eq. 9 to the flow rate is therefore in the order of 0.01% and we neglect it as well. The time derivative of VH(t) is given as

(10)dVH(t)dt=βVH0dTRdt. (10)

Now we proceed with the first term of Eq. 5. We denote the temperature measured inside the reservoir as TM(t). The temperature field in water inside the system can be expressed as T(x, t)=TM(t)+ΔT(x, t). The water density ρ(x, t)=ρ(TM(t)+ΔT(x, t)) can be expanded to linear order in ΔT(x, t), and the time derivative can be performed on this expansion. In this way, we obtain

(11)VH(t)ρ(x,t)tdV=ρ(TM(t))TdTM(t)dt.VH(t)+2ρ(TM(t))T2.dTM(t)dtVH(t)ΔT(x,t)dV+VH(t)ΔT(x,t)tdV, (11)

where the first term is the volume expansion based on the measured temperature TM. The second term is a correction for a non-homogenous temperature (differences in temperature lead to variations in the thermal expansion coefficient), whereas the third term is a correction for a non-homogeneous temperature increase rate. Higher-order terms have been neglected. The integral in the second term of 5 can be rewritten as

(12)VH(t)ΔT(x,t)dV=VH(t)·(TA(t)-TM(t)), (12)

where TA(t) is the average water temperature in the heated part of the system. The difference of the average temperature and the measured temperature is denoted c(t), i.e.

(13)c(t)=TA(t)-TM(t). (13)

The properties of the function c(t) are described in the section Uncertainty Budget, where the function is investigated numerically. The integral in the third term of Eq. 5 can be rewritten as

(14)VH(t)ΔT(x,t)tdV=VH(t)dcdt+dVHdt(TA-T(VH)). (14)

The second term of Eq. 14 can be written as βVH0(dTR/dt)(TA-T(VH)). If we compare the contribution of this term to the flow rate with the leading term (the first term in Eq. 5), we see that this term is lower than the leading term by a factor of approximately β(TA-T(∂VH)). The value TA-T(∂VH) does not exceed 1 K in our system, and β is in the order of 10-5/K. Therefore, the contribution of the second term of Eq. 5 is 5 orders lower than the leading term in the worst case. This term is therefore negligible and we do not keep it further.

Combining all equations above, we obtain the following expression for the flow rate:

(15)Q=-VH0ρ(TMUT)[(1+β(TR(t)-T0)){ρ(TM)T(dTMdt+dc(t)dt)+2ρ(TM)T2dTMdtc(t)}+ρ(TR)βdTRdt]. (15)

See the section Theoretical Model for the nomenclature. The partial derivatives follow the Tanaka or IAPWS 1995 equations (Tanaka equations valid up 40°C), given below:

(16)ρIAPWS=co1+c1Tn+c2Tn2+c3Tn31+c4Tn+c5Tn2, (16)

where ρIAPWS (kg/m3) is the density according to the IAPWS formulation [3] for air-free water at 101,325 Pa, Tn is the normalized temperature (T/100), and T is the temperature (°C). The coefficients are given by c0=999.84382, c1=1.4639386, c2=-0.015505, c3=-0.0309777, c4=1.4572099, and c5=0.0648931.

(17)ρTanaka=α5(1-(T+α1)2(T+α2)α3(T+α4)), (17)

where ρTanaka (kg/m3) is the density according to the Tanaka formulation [20] for air-free water at 101,325 Pa and T is the temperature (°C). The coefficients are given by α1=-3.983035, α2=301.797, α3=522528.9, α4=69.34881, and α5=999.974950.


[1] Ahrens M, Klein S, Nestler B, Damiani C. Design and uncertainty assessment of a setup for calibration of microfluidic devices down to 5 nl min-1. Meas Sci Technol 2014; 25: 015301.10.1088/0957-0233/25/1/015301Search in Google Scholar

[2] Ahrens M, Damiani C, Klein S, et al. An experimental setup for traceable measurement and calibration of liquid flow rates down to 5 nl/min. Biomed Eng-Biomed Tech 2015; 60: 337–347.10.1515/bmt-2014-0153Search in Google Scholar PubMed

[3] Batista E, Paton R. The selection of water property formulae for volume and flow calibration. Metrologia 2007; 44: 453–463.10.1088/0026-1394/44/6/004Search in Google Scholar

[4] Beek MP van der, Lucas P. Realizing primary reference values in the nanoflow regime, a proof of principle. Meas Sci Technol 2010; 21: 074003.10.1088/0957-0233/21/7/074003Search in Google Scholar

[5] Bissig H, Petter HT, Lucas P, et al. Primary standards for measuring flow rates from 100 nl/min to 1 ml/min – gravimetric principle. Biomed Eng-Biomed Tech 2015; 60: 301–316.10.1515/bmt-2014-0145Search in Google Scholar PubMed

[6] Bronkhorst datasheet μ-Flow, Series L01/L02 digital mass flow meters/controllers for liquids, 2014. Available at: in Google Scholar

[7] Cox MG. Evaluation of key comparison data. Metrologia 2002; 39: 589–595.10.1088/0026-1394/39/6/10Search in Google Scholar

[8] Cox MG. The evaluation of key comparison data: determining the largest consistent subset. Metrologia 2007; 44: 187–200.10.1088/0026-1394/44/3/005Search in Google Scholar

[9] European Medicines Agency 2013 annex I summary of product characteristics. Prialt-EMEA/H/C/000551-IB/0039.Search in Google Scholar

[10] Graham E, Glenn N. Assessment of calibration and traceability requirements for ultra-low flow rates. Technical report NEL 2007239.Search in Google Scholar

[11] Haneveld J, Lammerink TSJ, de Boer MJ, et al. Modeling, design, fabrication and characterization of a micro Coriolis mass flow sensor. J Micromech Microeng 2010; 20: 125001.10.1088/0960-1317/20/12/125001Search in Google Scholar

[12] Hasselberg R, de Jong GJ, Somsen GW. Low-flow sheathless capillary electrophoresis-mass spectrometry for sensitive glycoform profiling of intact pharmaceutical proteins. Anal Chem 2013; 85: 2289–2296.10.1021/ac303158fSearch in Google Scholar PubMed

[13] Heemskerk AA, Busnel JM, Schoenmaker B, et al. Ultra-low flow electrospray ionization-mass spectrometry for improved ionization efficiency in phosphoproteomics. Anal Chem 2012 84: 4552–4559.10.1021/ac300641xSearch in Google Scholar PubMed

[14] International Organization for Standardization (Geneva). Guide to the expression of uncertainty in measurement, 1993. in Google Scholar

[15] ISO 2010 infusion equipment for medical use. ISO 8536-1 to 12.Search in Google Scholar

[16] Kuo JTW, Yu L, Meng E. Micro-machined thermal flow sensors – a review. Micromachines 2012; 3: 550–573.10.3390/mi3030550Search in Google Scholar

[17] Lucas P (coordinator). Metrology for Drug Delivery. EU-funded research project, 2012–2015. in Google Scholar PubMed

[18] Sensirion datasheet LG16-0025. Liquid mass flow meter, 2012.Search in Google Scholar

[19] Sparreboom W, van Geest J, Katerberg M, et al. Compact mass flow meter based on a micro Coriolis flow sensor. Micromachines 2013; 4: 22–33.10.3390/mi4010022Search in Google Scholar

[20] Tanaka M, Girard G, Davis R, et al. Recommended table for the density of water between 0°C and 40°C based on recent experimental reports. Metrologia 2001; 38: 301–309.10.1088/0026-1394/38/4/3Search in Google Scholar

[21] Wirth T. Microreactors in organic chemistry and catalysis, 2nd edition. New York: Wiley, 2013.10.1002/9783527659722Search in Google Scholar

[22] Zhou F, Lu Y, Ficarro SB, et al. Nanoflow low pressure high peak capacity single dimension LC-MS/MS platform for high-throughput, in-depth analysis of mammalian proteomes. Anal Chem 2013; 84: 5133–5139.10.1021/ac2031404Search in Google Scholar PubMed PubMed Central

Received: 2014-10-19
Accepted: 2015-6-26
Published Online: 2015-8-5
Published in Print: 2015-8-1

©2015 by De Gruyter