# Abstract

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.

# Acknowledgments

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 *ρ*_{MUT} of the water just upstream of the MUT. The volumetric flow rate through the MUT is then given by

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

where ∂*V*(*t*) is the boundary surface of the volume *V*(*t*) and *V*(*t*) is the boundary surface between water and titanium and *V*(*t*) into two parts: one part is denoted *V*_{H}(*t*) and has a positive temperature increase rate (heated fluid elements); the second one is denoted *V*_{C}(*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 ∂*V*_{H}(*t*) and ∂*V*_{C}(*t*). Therefore, we have

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 ∂*V*_{H}(*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

The volume of the heated part of the system *V*_{H}(*t*) is given by its measured reference value for certain temperature *T*_{0} 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 *T*_{R}(*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 *β*_{T} and *β*_{S} as the volume thermal expansion coefficients of the titanium and the stainless steel both at the temperature *T*_{0}. We obtain

where *V*_{T}(*T*_{0}) and *V*_{S}(*T*_{0}) are the volumes of water-filled cavities inside the titanium reservoir and inside the stainless steel capillary at temperature *T*_{0}. These volumes will be known from differential mass measurements. *T*_{R}(*t*) is measured by thermometer installed in a copper mounting attached to the titanium part. Eq. 7 can be rewritten as

where *T*_{0} and *β* is given by

The difference *β*_{S}-*β*_{T} is of the same order of magnitude as *β*_{T}, 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 *V*_{H}(*t*) is given as

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

where the first term is the volume expansion based on the measured temperature *T*_{M}. 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

where *T*_{A}(*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.

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

The second term of Eq. 14 can be written as *β*(*T*_{A}*-T*(∂*V*_{H})). The value *T*_{A}-*T*(∂*V*_{H}) 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:

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:

where *ρ*_{IAPWS} (kg/m^{3}) is the density according to the IAPWS formulation [3] for air-free water at 101,325 Pa, *T*_{n} is the normalized temperature (*T*/100), and *T* is the temperature (°C). The coefficients are given by *c*_{0}=999.84382, *c*_{1}=1.4639386, *c*_{2}=-0.015505, *c*_{3}=-0.0309777, *c*_{4}=1.4572099, and *c*_{5}=0.0648931.

where *ρ*_{Tanaka} (kg/m^{3}) 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 *α*_{1}=-3.983035, *α*_{2}=301.797, *α*_{3}=522528.9, *α*_{4}=69.34881, and *α*_{5}=999.974950.

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**Received:**2014-10-19

**Accepted:**2015-6-26

**Published Online:**2015-8-5

**Published in Print:**2015-8-1

©2015 by De Gruyter