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

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

(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=ddt∫V(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 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

(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 ∂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 ρ∂VH(t). Hence, the second term of Eq. 5 can be written as

(6)∫∂VH(t)ρ(x, t)ξ→(x, t)dS→=ρ∂VH(t)∫∂VH(t)ξ→(x, t)dS→ =ρ∂VH(t).dVH(t)dt. (6)

The volume of the heated part of the system V_H(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 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 ρ∂VH(t)=ρ(TR). Next, we denote β_T and β_S 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 V_T(T₀) and V_S(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. T_R(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 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

(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 T_M(t). The temperature field in water inside the system can be expressed as T(x, t)=T_M(t)+ΔT(x, t). The water density ρ(x, t)=ρ(T_M(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)dt∫VH(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 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

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

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.

(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(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:

(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 ρ_IAPWS (kg/m³) 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 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 ρ_Tanaka (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.