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.
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.
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
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 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 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
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 Hence, the second term of Eq. 5 can be written as
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 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
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
where is the total reference volume at T0 and β is given by
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
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
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
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.
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 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:
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 ρIAPWS (kg/m3) is the density according to the IAPWS formulation  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.
where ρTanaka (kg/m3) is the density according to the Tanaka formulation  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.
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