A new method for flow measurement in cryogenic systems

Cryogenics 60 (2014) 9–18

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Cryogenics journal homepage: www.elsevier.com/locate/cryogenics

A new method for flow measurement in cryogenic systems S. Grohmann ⇑ Karlsruhe Institute of Technology, Institute for Technical Thermodynamics and Refrigeration, Kaiserstr. 12, 76131 Karlsruhe, Germany Karlsruhe Institute of Technology, Institute for Technical Physics, Hermann–von-Helmholtz–Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

a r t i c l e

i n f o

Article history: Received 19 September 2013 Received in revised form 5 January 2014 Accepted 11 January 2014 Available online 23 January 2014 Keywords: Measurement Flowmeter Sensor Calibration Heat exchanger

a b s t r a c t A new method for mass flow measurement of fluids in pipes is presented; its novelty lies in the capability for intrinsic calibration. The method is founded on a concept, where two independent analytic expressions for the flow rate are formed from the same direct measurement readings (input parameters). If the input parameters were error-free, the two expressions would yield identical results, by definition. This fact can be used as goal function in a minimization routine that removes systematic errors of the inherently error-prone input parameters. The uncertainty of the mass flow measurement is then only influenced by statistical effects and is typically less than 1% with regard to the measured value. The new method is explained by a proof-of-principle that is based on measurements in a large-scale cryogenic system. The intrinsic calibrations can be executed in situ at any moment during operation of a plant, and with no need for a reference standard. While the new method is applicable in any system involving single-phase fluid flow, it offers particular advantages in cryogenic application. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Flow measurement is a standard task in any technical application involving fluid flow. A large variety of methods is known for the measurement of gas flow in pipes, such as differential pressure, vortex, ultrasonic, Coriolis and thermal flow measurement. In thermal or caloric mass flow meters, heat is added in heating elements and heat transfer functions to the fluid are evaluated. The traditional two-element principle shown in Fig. 1 comprises two consecutive elements in flow direction, which are electrically heated and cooled by the fluid through the conductive wall. An element may consist of a platinum wire, to which an electrical current is applied for heating and where the voltage drop is evaluated for temperature measurement at the same time. If both elements are heated with the same heat load Q_ while the fluid is stagnant, the temperature profile in the tube wall is symmetric with regard to the median between the elements, and the temperature difference DT between the elements is theoretically zero (dotted line in Fig. 1). In case of fluid flow, the temperature profile is shifted in flow direction and a temperature difference DT – 0 can be measured that is proportional to the mass flow rate (full line). In such thermal systems, the correlation between the tempera_ is complex. It is influture difference DT and the mass flow rate m enced by design (i.e. distance between elements, tube size,

⇑ Address: Karlsruhe Institute of Technology, Institute for Technical Thermodynamics and Refrigeration, Kaiserstr. 12, 76131 Karlsruhe, Germany. Tel.: +49 72160842332. E-mail address: [email protected] 0011-2275/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cryogenics.2014.01.004

materials and shape of heat exchangers – affecting axial and radial thermal resistances, as well as contact resistances), by flow and fluid parameters (i.e. viscosity, conductivity, specific heat), by sensor orientation and by surrounding conditions. A large number of design solutions1 is known that aim for the minimization of error effects and usually for linear sensor characteristics. The functional cor_ ¼ f ðDTÞ, however, can only be determined by multi-point relation m calibration and stored in form of sensor-specific curve coefficients. The latter generally holds for all other measuring principles, all involving models with empiric components that must be determined by calibration in test stands under reference conditions. Apart from a solution [1], no flow meter for cryogenic application can be found as serial product on the market today, mainly because a manufacturer calibration with low-temperature helium or hydrogen is rather unfeasible in terms of cost and effort. In this paper, a new method for flow measurement is presented, with the ability for intrinsic calibration that can be executed during operation of a cryogenic installation. The intrinsic calibration is founded on a new concept, whereby two independent analytic expressions for the flow rate are formed from the same measurement readings (input parameters). If the input parameters were error-free, the two expressions would yield identical results, by definition. This fact can be used as goal function in a minimization routine that removes systematic errors from the inherently error-prone input parameters. The standard uncertainty of the flow rate is then only

1 In the three-element principle, for instance, the heating and temperature measurement functions are separated.

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S. Grohmann / Cryogenics 60 (2014) 9–18

Nomenclature A AW C_

reference surface of the heat exchanger inner tube surface capacitance flow specific heat capacity of the fluid inner tube diameter systematic error of parameter i overall heat transfer coefficient w.r.t. A heat exchanger length dimensionless length parameter mass flow rate Nusselt number Prandtl number heat flow thermal resistance of the heat exchanger contact resistance virtual resistance

cp d Fi k L e L _ m Nu Pr Q_ R RC RV

Element 1

Ra Rk Re TA T 0F T 00F DT DT 0 DT 00 DT m DT sat u w

convective resistance conductive resistance Reynolds number temperature of the heat exchanger surface A fluid inlet temperature fluid outlet temperature temperature difference inlet temperature difference outlet temperature difference logarithmic mean temperature difference saturation temperature shift standard uncertainty fluid velocity convective heat transfer coefficient fluid bulk viscosity fluid viscosity at wall temperature

a g gw

Element 2

. Q

Bypass

TA

TF´

Pipe

d

d

Laminar flow element

z

TF´´

Heat exchanger

T

Fig. 2. General principle of the sensor design.

Stagnant fluid

Flowing fluid

ΔT

x Fig. 1. Measuring principle of thermal flow meters.

influenced by random errors and is typically um_ < 1% with regard to the measured value.2 The new method is introduced in Section 2 with the aim of first presenting a global picture of the underlying model. For the sake of clarity, no details are discussed at this point on how the model conditions can be implemented. The method’s feasibility is proven in the subsequent Section 3 on the basis of experimental data. Two options for transient flow measurement are presented in Section 4, as the intrinsic calibration requires quasi-static flow conditions. Design features and peculiarities of operation are then discussed in Section 5, explaining why this method works even under practical conditions that deviate to some extend from the analytic model. Conclusions are drawn in the final Section 6 together with an outlook for future activities. 2. The new measuring method 2.1. General layout and expressions The new method proposed in this paper is a thermal method, yet it is fundamentally different from all thermal flow measuring 2 The uncertainty of commercial flow meters is usually given in percent of the measurement range.

principles known so far.3 The new method is based on the general layout of fluid flow through a heat exchanger as illustrated in Fig. 2. The heat exchanger is heated with a controllable heat load Q_ and is designed in such a way that the surface temperature T A is nearly constant in flow direction z. Upstream and downstream of the heat exchanger, the fluid inlet and outlet temperatures T 0F and T 00F are measured, respectively. In contrast to classical thermal flow meters, T 0F and T 00F are installed in such distances from the heat exchanger, where:  radial temperature profiles in the fluid are negligible,  the temperature rise in the tube wall by axial heat conduction from the heat exchanger is negligible, and  no requirements on the symmetry of T 0F and T 00F with regard to the heat exchanger exist. The measurement readings of Q_ ; T A ; T 0F and T 00F can be reduced to three measuring quantities, expressing the temperature measurements in form of inlet and outlet temperature differences DT 0 and DT 00 as shown in Fig. 3. With the three quantities Q_ ; DT 0 and DT 00 , two analytic expressions can be formed. The first expression is the energy balance of the fluid flow4

  _ cp DT 0  DT 00 Q_ ¼ m

ð1Þ

where cp is the specific heat capacity of the fluid. The second expression is given by kinetics in the heat exchanger

3 Common features in all thermal methods are the required knowledge of the fluid’s specific heat capacity cp and the restriction to single-phase flow. 4 The constant surface temperature T A cancels out in (1), hence ðDT 0  DT 00 Þ ¼ ðT 00F  T 0F Þ.

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S. Grohmann / Cryogenics 60 (2014) 9–18

case of liquid flow, scaling with the fluid’s viscosity ratio at bulk and wall temperatures

Nu / ðg=gW Þ0:14

TF´ z Fig. 3. Schematic view of temperature profiles in the heat exchanger along the flow direction z. 0

1 DT  DT  0 Q_ ¼ k A DT m ¼ R ln DDTT00

00

ð2Þ

Here, k is the overall heat transfer coefficient with regard to the reference surface A of the heat exchanger. The embedded logarithmic mean temperature difference

DT m ¼

DT 0  DT 00  0 ln DDTT00

ð3Þ

is an analytic expression; for its derivation see e.g. [2]. The reciprocal of ðk AÞ is equivalent to the overall thermal resistance R of the heat exchanger. From (1) and (2), two analytic expressions can be formed for the _ The first expression m _ A is obtained simply by mass flow rate m. rearranging (1)

_A¼ m

Q_  cp DT  DT 00 

ð4Þ

0

_ B is obtained by equating (1) and (2) and The second expression m _ solving for m

_B¼ m

1  0 R cp ln DDTT00

ð5Þ

One may wonder whether (4) and (5) are really independent, as (5) was derived by equating (1) and (2). Mathematically, they are linearly independent, because (5) cannot be expressed as a linear combination of (4). 2.2. The thermal resistance R The overall thermal resistance R of the heat exchanger results from the sum of resistances Ri . This is illustrated in Fig. 4 for a simple design, where e.g. a conductive metal block is brazed onto a stainless steel tube. The conductive resistances Rk;i of the material layers, on the one hand, can be considered constant at plus/minus a few kelvins around the actual operating temperature. The same applies for the contact resistances RC;j . The convective heat transfer resistance Ra , on the other hand, is a function of the Nusselt number Nu. Nu depends on Reynolds ðReÞ and Prandtl ðPrÞ numbers and can be expressed as

Nu ¼ f ðRe; PrÞ ¼ f ðw; fluid propertiesÞ

ð6Þ

where w is the flow velocity of the fluid. While (6) generally holds for gas flow, Nu is additionally affected by the wall superheating in RC,1 = const. RC,2 = const.

. Q

Rλ,1 = const. Rλ,2 = const. Rλ,3 = const. Rα = f (Nu)

Fig. 4. Parameters inuencing the overall thermal resistance R of the heat exchanger (cross-sectional view).

ð7Þ

(see e.g. [3]). The effect is small, however, accounting for c. 103 per kelvin temperature difference between the inner tube surface and the bulk fluid in case of water. It is obvious that R cannot be calculated with the precision needed for a sensor. However, R can be determined experimentally _ and variable heat with measurements at constant mass flow rate m load Q_ . Within a few kelvins of temperature change DT due to the variable heat load, the thermal resistance R remains constant because of

ðw; fluid propertiesÞm_ ¼ const:

ð8Þ

,!Ra ¼ const: ðmaterial propertiesÞTDT ¼ const:   ,! Rk;i ; RC;j TDT ¼ const: X X R ¼ Ra þ Rk;i þ RC;j ¼ const: In such data series with a stepwise change of Q_ ; R is the slope of a function

DT m ðDT 0 ; DT 00 Þ ¼ R Q_ ðþzero pointÞ

ð9Þ

thus R can be obtained by linear regression. Experimental example data for such a procedure are shown in Fig. 5.

2.3. Influence of measuring errors The measurement readings of Q_ ; DT 0 and DT 00 are affected inher_ 1 ¼ 1 g=s heently by measuring errors. With example data of m lium flow rate at T 0F ¼ 100 K freely chosen for illustration, Fig. 6 gives an impression on how even small errors fF Q_ ; F DT 0 ; F DT 00 g can _ A ¼ f ðQ_ Þ and lead to strong systematic effects of the functions m _ B ¼ f ðQ_ Þ. For the chosen theoretical data and four arbitrary comm binations of fF Q_ ; F DT 0 ; F DT 00 g it is shown that the systematic effects of the two functions may yield (a) (b) (c) (d)

opposite trends, both positive or both negative deviations, or even sign inversions 4

Measuring data Linear fit

3

ΔTm (K)

TF´´

ΔT ´´

ΔT ´

T

TA

2

1

0

0

2

. Q (W)

4

6

Fig. 5. Determination of the thermal resistance R by linearregression of measured _ = const. and variable Q_ , using (9) as fit function. data at m

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S. Grohmann / Cryogenics 60 (2014) 9–18

100

100

. . Δm / m (%)

200

. . Δm / m (%)

200

0

-100

0

-100

(a) -200

-2

0

2

. Q (W)

4

(b) -200

6

100

100

. . Δm / m (%)

200

. . Δm / m (%)

200

0

-100

-2

0

2

4

. Q (W)

0

-100

(c) -200

6

-2

0

2

. Q (W)

4

(d) 6

-200

-2

0

2

4

. Q (W)

6

_ A ðNÞ and m _ B ðMÞ at constant ow rate and variable heat load due to errors of the measurement readings of Q_ ; DT 0 and DT 00 . Fig. 6. Systematic effects of the functions m _ 1 ¼ 1 g=s ¼ const:; Q_ ¼ f1; 2; 3; 4g W; T 0F ¼ 100 K; cp ¼ 5:2 kJ=kg K; R ¼ 0:3 K=W. Assumed systematic errors of the measuring quantities Theoretical example data (): m fF Q_ ; F DT 0 ; F DT 00 g in units of (W, K, K): f0:868; 0:046; 0:026gaÞ ; f0:387; 0:103; 0:008gbÞ ; f1:446; 0:113; 0:082gcÞ ; f0:061; 0:123; 0:125gdÞ .

_ ¼ ðm _ fA;Bg  m _ 1 Þ. Hereby, the magnitude of deviations can of Dm easily reach plus/minus several hundred percent, which demonstrates why the measurement in thermal mass flow meters is usually not just based on the fluid’s energy balance (4).

Let us assume that the fluid’s heat capacity cp is known and that the direct measurement readings Q_ ; DT 0 and DT 00 are recorded at some unknown flow rate. These readings can be evaluated in two ways in order to obtain the mass flow rate:  On the one hand, the measurement readings can be evaluated _A¼m _A by (4), hence the first evaluation function is m ðQ_ ; DT 0 ; DT 00 Þ.  On the other hand, the same measurement readings can be eval_B¼m _ B ðR; DT 0 ; DT 00 Þ. At uated by (5) with dependencies of m _ ¼ const:; R can be obtained from the linear fit function (9), m where R ¼ RðQ_ ; DT 0 ; DT 00 Þ. Plugging (9) into (5) yields the same input parameters for the second evaluation function, namely _B¼m _ B ðQ_ ; DT 0 ; DT 00 Þ. m _ A ðQ_ ; DT 0 ; DT 00 Þ and m _ B ðQ_ ; DT 0 ; The two evaluation functions m DT 00 Þ receive inherently error-prone input parameters, i.e. the direct measurement readings. The error-prone input parameters, however, can be turned into error-free parameters following the principle

x  F x ¼ x

ð10Þ

where F x is the error5 of x, and x is error-free. Substituting all input parameters by (10) yields two analytic functions with error-free parameters, which must yield identical results by definition 5 We may first think of an overall error F x . The distinction between systematic and statistical error components is discussed later.

ΔTm,i – ΔT m,fit (K)

2.4. Analytic principle

0.010

0.005

0 1

2

3

4

i (–)

–0.005

–0.010 Fig. 7. Residuals of the data in Fig. 5. The residuals are randomly scattered, confirming that the linear fit model (9) was appropriate. Based on the residuals, _ ¼ const. criteria may be programmed for automatic rejection or approval of m

_ A ðQ_  F Q_ ; DT 0  F DT 0 ; DT 00  F DT 00 Þ ¼! m _ B ðQ_  F Q_ ; DT 0  F DT 0 ; DT 00  F DT 00 Þ m

ð11Þ

Eq. (11) is the core of the new method. It is used as goal function in a minimization routine, with the aim of removing the systematic ef_ A and m _ B. fects from both evaluation functions m 2.5. Algorithm for determining the mass flow rate The following algorithm is used in order to determine the mass _ on the basis of (11). flow rate m Step A Measure a series with n P 2 measuring points at constant flow _ Each measuring point may consist of several readings of rate m. fQ_ ; DT 0 ; DT 00 g with Q_ ¼ const:, but only the mean values are needed

13

S. Grohmann / Cryogenics 60 (2014) 9–18

in the subsequent evaluation.6 Different measuring points are achieved by ramping the heat load Q_ iþ1 – Q_ i . The minimum of nmin ¼ 2 measuring points is required for the linear regression of R. More than 2 measuring points allow for a _ was indeed constant, assessing the residuverification whether m als with regard to the linear fit (cf. Fig. 7). This is possible because R _ ¼ const: Hence, the linearity of can only be constant if fw; mg _ B , and the DT m ¼ f ðQ_ Þ is both a necessity for the evaluation of m _ ¼ const: In case of m _ – const:, reject the data verification of m and try again.

GNe inlet

Connection to external Ne buffer

165 mm

He outlet

Step B Extend the mean values of fQ_ i ; DT 0i ; DT 00i gi¼1;...;n by their corresponding error parameters F Q_ ; F DT 0 and F DT 00 in accordance with (10) and send them to the evaluation functions

Outer cylinder with helical groove

Lead core

Outer wall (shrunk-fit) He inlet LNe outlet 84 m m

_ A ðQ_ i  F Q_ ; DT 0i  F DT 0 ; DT 00i  F DT 00 Þ m i¼1;...;n _ B ðQ_ i  F Q_ ; DT 0i  F DT 0 ; DT 00i  F DT 00 Þ m i¼1;...;n

Fig. 8. Neon condenser in the tritium source of KATRIN [4].

Typically, the error parameters may have initial values of fF Q_ ; F DT 0 ; F DT 00 g ¼ 0. For n ¼ 4 measuring points, one thus obtains 4 mass flow _ A and 4 mass flow rates from m _ B . Then combine those rates from m mass flow rates to one data set, consisting in this example of 2n ¼ 8 values for the flow rate.

_ A;fit and m _ B;fit are equal, their results must be exact within If m the limits of their random errors.7 The logic of the algorithm consequently yields an intrinsic calibration of the constant mass flow rate under which the data were taken. 3. Proof-of-principle The experimental data presented hereafter were obtained during tests of the beam tube cooling system for the tritium source in KATRIN [5]. The cooling system consisted of a two-phase thermosiphon operated at 30 K, where neon was evaporated by the heat load on the beam tube, and re-condensed by gaseous helium with T 0F  26 K. The neon condenser shown in Fig. 8 represented a 6 At 1 kHz sampling rate, for instance, 1000 readings of Q_ ; DT 0 and DT 00 could be recorded in one second and the mean values calculated. For the sake of simplicity, such mean values are not separately marked with overlines in this paper. 7 _ A; m _ B gi in just one measuring point i is not sufficient and can be The equality of fm coincidental, as e.g. illustrated by the curve crossing in Fig. 6(d). This does not conflict _ B is undefined in just one measuring point. with this statement, because m

. mA (raw data) . mB (raw data) . mA,fit . mB,fit

6

. mHe (g/s)

Step C With Step B programmed as a function, minimize the standard deviation of the data set by variation of the free (error) parameters fF Q_ ; F DT 0 ; F DT 00 g. For consistency, the error parameters fF DT 0 ; F DT 00 g are also passed inside the linear regression function, whereas F Q_ does not affect the slope of (9). The minimization procedure is a flexible way of implementing _A¼m _ B gi defined in (11). Since fF Q_ ; F DT 0 ; F DT 00 g signify the goal of fm only systematic error components, a remaining inequality is possi_ A; m _ B gi in each of the n measuring points due to stable between fm tistic (random) errors. ‘‘Systematic’’ implies that the same values of fF Q_ ; F DT 0 ; F DT 00 g are subtracted from each fQ_ i ; DT 0i ; DT 00i gi¼1;...;n , both in _ A and in m _ B. m The minimization of the standard deviation pushes the calculated mass flow rates toward equality with no further limitation. _ A and m _ B must be The underlying goal function (11) states that m equal, if the input parameters were error-free. Error-free input _ A; m _ B g and hence, the inversion parameters yield exact results of fm of the argument implies:

8

4

2

0

0

2

4

6

. Q (W)

8

10

12

Fig. 9. Determination of the helium ow rate using the procedure in Section 2.5. Fitted _ He ¼ 2:049 g=s; remaining standard uncertainty um_ He ¼ 0:005 g=s (0:2% mean value m of measured value).

heat exchanger with constant surface temperature on the inner tube (i.e. the saturation temperature of neon), and with helium gas flow in the outer shell. The helium temperatures were measured with TVO sensors installed on the connecting tubes upstream of the ‘‘He inlet’’ and downstream of the ‘‘He outlet’’. The saturation temperature of neon was calculated from saturation pressure measurements using REFPROP [6]. For obtaining the data presented in Fig. 9, the heat load in the neon circuit was ramped using electrical heaters. Each of the n ¼ 4 measuring points consisted of c. 1000 data records taken over 30 min of stationary operation,8 from which mean values and standard deviations were calculated. The stability of the helium flow rate could be verified by Figs. 5 and 7, which were created from these mean values. The evaluation of the raw data with (4) and (5) yielded strong _ A ðNÞ and m _ B ðMÞ in Fig. 9. The syssystematic effects, indicated by m _ A and m _ B with regard to the average values tematic deviations of m of each 4 results was within f3; . . . ; 45%g. The combined standard uncertainties calculated from purely statistical variations of the c. 1000 records in each measuring point were within f20; . . . ; 65%g (error bars in Fig. 9). 8 The installation was not supposed to be a flow sensor, where it is unnecessary to take so many data over such a long period. The original purpose was rather to investigate the stability of the cooling system (see [5]).

14

S. Grohmann / Cryogenics 60 (2014) 9–18

Processing the mean values fQ_ i ; DT 0i ; DT 00i g of the i ¼ 1; . . . ; 4 measuring points with Steps B and C of Section 2.5 yielded _ A;fit ðþÞ and m _ B;fit ðÞ, showing clearly that the systematics were m fully removed from the two functions. The horizontal shift was mainly caused by the static heat load on the beam tube, being included in the ‘‘systematic error’’ F Q_ . Evaluating the algorithm in MathematicaÒ [7] together with REFPROP [6] on a PC took only _ A;fit ; m _ B;fit g yielded a helium flow rate 0.2 s. The mean value of fm _ He ¼ 2:049 g=s. of m _ A;fit ðþÞ did not match As shown more clearly in Fig. 11(d), m _ B;fit ðÞ in each of the 4 measuring points due to random precisely m _ A;fit ; m _ B;fit g errors. Hence, the standard deviation of the 8 results fm could be regarded as the actual standard uncertainty of the intrinsic mass flow calibration. It yielded um_ He ¼ 0:005 g=s in this example, corresponding to 0:2% with regard to the measured value. The method was also applied to further measurements in the same installation, yielding helium flow rates of

_ He ¼ ð0:675  0:007Þ g=s m _ He ¼ ð1:095  0:006Þ g=s m Here, one has to emphasize that these results were not obtained with a specially designed and tuned sensor, but rather with a heat exchanger in a large and very stable cryostat, and with PLC-controlled process instrumentation. The introduction of the new method with its capability for intrinsic calibration certainly asks for comparative measurements against a standard. At the moment, such comparative results cannot be provided, because the neon condenser was installed in one of three parallel branches,9 and only the total helium flow rate was measured _ He;total  7 g=s). Notice, however, that the model with a Venturi tube (m has no empiric components and consists ‘‘only’’ of two analytic functions. Once such two functions exist, the minimization routine cannot fail to deliver the correct flow rate (cf. Step C). While the first function _ A (4) is straightforward, it is crutial for the second function m _ B (5) to m _ Therehave a heat exchanger with a linear dependence DT m ¼ f ðQÞ. fore, the proof of linearity by Figs. 5 and 7 might be accepted as a proof-of-principle for the new method. Nevertheless, a new sensor for comparative measurements at room-temperature against a calibrated orifice is currently being prepared. The results will be published shortly. 4. Measurement of transient flow It has been demonstrated in Sections 2 and 3 how the mass flow rate can be determined by intrinsic calibration, ramping the heat load, waiting for equilibrium and calculating mean values at each step, verifying the constancy of the flow rate and minimizing the _ A and m _ B . The time needed for a meadispersion of results from m surement in such calibration mode depends on the time constant of the sensor, and is typically in the range of several seconds. During this period the mass flow must be constant, otherwise R cannot be _ B be evaluated. determined and m In cryogenic installations, operating conditions are often quasistationary and hence, the mass flow rate might always be measured in calibration mode. Nevertheless, transient measurement capabilities will also be required, for instance to provide a continuous (analog) sensor signal. Both a theoretical and a practical option are presented hereafter. 4.1. Theoretical option

the measurement readings fQ_ ; DT 0 ; DT 00 g could be corrected instantly, and the energy balance (4) applied solely for measurements in transient mode. The implementation in practice, however, will be challenging since it requires the constancy of fF Q_ ; F DT 0 ; F DT 00 g even under changing flow and ambient conditions. Independent of its potential for practical realization, the theoretical option is explained, because it provides some deeper insight in the capabilities of the new method. From a mathematical point of view, the algorithm in Section 2.5 is characterized by: _ A and m _ B ),  2 equations (i.e. m  3 unknowns (i.e. F Q_ ; F DT 0 ; F DT 00 ). This implies an under-constrained system of equations, yielding in principle an infinite number of solutions fF Q_ ; F DT 0 ; F DT 00 g. As long _ is to be extracted from the minimization routine in calas only m ibration mode measurements, the actual values of fF Q_ ; F DT 0 ; F DT 00 g do not matter at all; any parameter combination at the fitted minimum is fine. However, such an arbitrary solution for _ A or m _ B for fF Q_ ; F DT 0 ; F DT 00 g only yields correct results from either m the very conditions, under which the calibration was executed. At any different flow rate, the functions

_ A; m _ B g ¼ f ðQ_  F Q_ ; DT 0  F DT 0 ; DT 00  F DT 00 Þ fm would exhibit systematic effects again. This behavior can be exploited for finding the exact solutions for fF Q_ ; F DT 0 ; F DT 00 g, evaluating two data series at arbitrary (but different) mass flow rates in common. The second data series then provides the third independent relation, which is needed for the exact determination of the 3 unknowns. To demonstrate the feasibility of such an extended model, a _ 2 with corresponding fQ_ i ; DT 0i ; DT 00i gi¼1;...;4 second mass flow rate m _ 1 already (cf. Table 1) was processed together with the data m discussed in Fig. 6. The purely theoretical data were altered with the assumed systematic errors of Fig. 6(a)–(d), and the standard _ A;1 ; m _ B;1 g and fm _ A;2 ; m _ B;2 g were minimized in one deviations of fm procedure.10 The numerical solutions are summarized in Table 1, showing that all given values of fF Q_ ; F DT 0 ; F DT 00 g could be reproduced by the minimization routine with a precision level of DF  105 . The times required by the solution algorithm were around 0:8 s. These results are interesting in several ways. First of all, the minimization routine proved to have no problem whatsoever in finding the correct minimum, even with extreme systematics as in Fig. 6(d). The large sensitivity of the evaluation functions (4) and (5) with regard to measuring errors fF Q_ ; F DT 0 ; F DT 00 g may be a problem when only one function is available. In the new method, however, this turns into an advantage, because it simplifies the convergence of the minimization routine. The large sensitivity provides further advantages, as a sensor can thus be operated with very small temperature differences of possibly DT  1 K. Such small temperature differences then enable operation, whereby  the sensor can be placed in the fluid’s main stream (at least up to certain pipe dimensions), so that bypass solutions as shown in Fig. 1 can be avoided,  heating of the fluid is small, which is essential in cryogenic application,  even liquid flow can be measured, because the remaining systematics of (7) is irrelevant for technical application.

So far, it has not been discussed whether the exact values of fF Q_ ; F DT 0 ; F DT 00 g can be determined. If those values were available, 9

Each branch was equipped with a control valve.

10 There are several mathematical options for a combined minimization of the standard deviations.

15

S. Grohmann / Cryogenics 60 (2014) 9–18

Table 1 _ 2 . The example data for both m _ 1 and the 4 combinations of _ 1 and m Precision of fitted systematic error parameters for the combined fitting of data from the mass flow rates m fF Q_ ; F DT 0 ; F DT 00 g are listed in Fig. 6. _ 2 ¼ 1:8 g=s; Q_ ¼ f1; 2; 3; 4g W; T 0F ¼ 100 K; cp ¼ 5:2 kJ=kg K; R ¼ 0:27 K=W m Case (b)

3:5  108

F DT 00 ;fit  F DT00

(K)

8

2:2  10

4.2. Practical option It is not clear to which level (and at which cost) the constancy of fF Q_ ; F DT 0 ; F DT 00 g can be achieved in a real sensor. Therefore, another option based on the conventional usage of characteristic maps looks more promising, whereby empirical functions are fitted through a number of sampling points. In this case, the capability for intrinsic calibration is a great advantage. Since calibrations can be executed in arbitrary sequence during operation, a truly adaptive sensor can be made. The results of each calibration can be stored in a database, providing an incremental mesh refinement of the sampling points. In this way, the measurement accuracy in transient mode can be improved progressively with the system’s operation. This practical option for transient mode measurements puts less constrains on the sensor design and has further advantages, which will be explored in the following Section. Details for its implementation are still under investigation. 5. Discussion of sensor design and operation _B 5.1. Validity of m _ B (5) is based on the kinetics of The second analytic function m heat transfer. The embedded expression DT m (3) is only correct, however, if the local heat transfer coefficient a is constant along the heat exchanger length z. Strictly, a – f ðzÞ could only be achieved with hydrodynamically and thermally developed flow. The latter condition cannot be met and hence there are higher values of a in the inlet as compared to the downstream area. However, we are not interested in local values of a, but rather in the average Ra ¼ 1=ða AW Þ, where AW is the inner tube surface. _ and What is important is that Ra does not change at constant m variable Q_ , which is confirmed by the empirical equation for thermally developing flow [2]

LÞ with Nute ¼ f ðe

e L¼

L d Re Pr

3:0  10

2:2  10

2:8  106

1:0  106

7:6  107

1:1  106

7

7

6:0  107

7:1  10

From a conceptual point of view, the experimental approach to compensate for missing equations could be used for further model extensions, i.e. for additional measuring quantities Y i with corresponding error parameters F yi . In order to determine the values of F yi ; i additional independent relations are then required. Those _ and relations could be provided by measurements at constant m variable Y i . One example may be the inclusion of the casing temperature in sensors for installation under ambient conditions in process plants. In cryostats, however, this will not be necessary.

ð12Þ

where L is the heat exchanger length and d the inner tube diameter. The ‘‘imperfection’’ of the heat exchanger with regard to the analytic model is then systematic and can be corrected. In fact, the minimization of the standard deviation is a ‘‘brute force’’ mathematical method, which does not care much about the origin of systematics. It removes any kind of systematics that can be removed.

Case (d) 6

5:3  10

5.2. Constant surface temperature T A The constant surface temperature T A is another issue that is difficult to achieve. It is actually impossible to design a heat exchanger, where T A – f ðzÞ holds for different flow rates and variable heat load. Even for the neon condenser shown in Fig. 8, the static vapor pressure head yielded a small saturation temperature difference of DT sat ¼ 0:6 mK over the heat exchanger height. Now, let us recall that the system of equations is under-constrained in calibration mode. For the minimization routine to converge, we need to set a boundary condition. Here, a limit of F DT;max can be chosen, signifying the range of values that F DT 0 and F DT 00 can assume, respectively. During the minimization, this boundary will be hit by either F DT 0 or F DT 00 . In principle, there are no physical restrictions in choosing values for the boundary. If F DT;max is increased beyond the maximum offset of the temperature measurements, virtual layers are added on the heat exchanger surface A, characterized by additional virtual offsets. This is explained by Fig. 10, showing a virtual resistance RV between the actual surface and a virtual surface. Increasing RV implies that the virtual surface temperature T A becomes more and more homogeneous, fulfilling the condition of the model. This is further illustrated by Fig. 11, where the experimental data of Section 3 were evaluated with 4 different values of F DT;max . The results in Fig. 11(a) correspond to F DT;max ¼ 0:10 K, which was approximately the uncertainty of temperature measurement _ A and m _B during those experiments. It can be noticed that both m have strong systematic effects remaining, indicating that there was ‘‘not enough room’’ to correct all systematics. The increase of F DT;max reduced the systematics, which were essentially gone at j F DT;max jP 0:16 K. For the results presented in Section 3 and in Fig. 11d), a boundary of F DT;max ¼ 0:50 K was chosen. Notice the value of F DT;max should not be exaggerated, as it reduces the _ B (5) and hence the sensitivity. slope of m

T

Virtual surface temperature TA°

FΔT ''

(K)

Case (c) 6

FΔT '

3:8  10

F DT 0 ;fit  F DT 0

8

RV

Actual surface ΣRλ wall e b u t r e n In Rα lk u b d Flui

ΔT '

(W)

temperature TA

ΔT ''

Case (a) F Q_ ;fit  F Q_

temperature t ur e tempera

TF''

TF' z Fig. 10. Schematic view of temperature profiles along the flow direction z in a real heat exchanger.

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S. Grohmann / Cryogenics 60 (2014) 9–18

The most beautiful property of RV is that it comes for free, while the heat exchanger design can be relatively simple. This degree of freedom, however, is not available with the extended model of Section 4.1. 5.3. Condensation heat transfer In the example of Section 3, heat was added to the helium flow by the condensation of neon. The stepwise change of the heat load implied a changing heat flux in the condenser, and consequently a changing condensation heat transfer coefficient. This violates the fundamental prerequisite R – f ðQ_ Þ of the new method. How does this fit together with the overall resistance R, which was definitely constant as proven by Figs. 5 and 7? The explanation is found in two reasons:  The first reason is that the dependence of condensation heat transfer on heat flux (or DTÞ is small, scaling with DT 0:25 .  The second reason is that the thermal resistance on the condensation side was only a small fraction of the overall resistance R. A small change of a small contribution yielded a very small effect, which just disappeared under the random noise.

5.4. Constancy of F Q_ As already mentioned in Section 3, the error parameter F Q_ does not only relate to the measurement of the heater power as such. It also includes any static amount of energy, which flows in form of heat to the ambience rather than to the fluid, or which is additionally absorbed from the ambience. Such forms of heat flow, e.g. by thermal conduction through sensor leads and by thermal radiation can never be avoided completely. Keeping track of those heat flows is unnecessary as long as F Q_ remains constant during ramping of Q_ in a measurement series of i ¼ 1; . . . ; n. This can be achieved by heater cabling with the 4wire technique typically used for temperature measurement. Changes of conductive or radiative losses/gains are negligible in view of the small temperature changes of the heater in a measurement series. 5.5. Fluid temperatures T 0F and T 00F In Section 2.1 it was mentioned that the fluid temperatures T 0F and T 00F are measured in places where radial temperature profiles in the fluid are negligible. This is straightforward for T 0F upstream of the heat exchanger, because no heat load implies no DT. There could be a small heat load from self-heating of a resistance 2.8

2.8

(a)

(b)

2.6

Mass flow (g/s)

Mass flow (g/s)

2.6

2.4

2.2

2.0

2.2

FΔT,max = 0.10 K . mHe = 2.444 g/s . = 4.7 % um,He 4

5

6

2.4

FΔT,max = 0.12 K . mHe = 2.268 g/s . = 2.6 % um,He

. mA,fit . mB,fit 7

8

2.0

9

4

5

Heat load (W)

6

. mA,fit . mB,fit 7

8

2.8

2.8

(c)

FΔT,max = 0.14 K . mHe = 2.114 g/s . = 0.8 % um,He

(d)

. mA,fit . mB,fit

FΔT,max = 0.50 K . = 2.049 g/s mHe . = 0.2 % um,He

2.6

Mass flow (g/s)

Mass flow (g/s)

2.6

2.4

. mA,fit . mB,fit

2.4

2.2

2.2

2.0

9

Heat load (W)

4

5

6

7

Heat load (W)

8

9

2.0

6

7

8

9

10

11

Heat load (W)

Fig. 11. Effects of different values of boundary conditions F DT ; max on results of the experimental data presented in Section 3. The boundary conditions affect the fitted values of F Q_ , hence F Q_ has no direct relation to the actual heat loss/gain.

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S. Grohmann / Cryogenics 60 (2014) 9–18

thermometer, but this again is a systematic effect that can be compensated. To avoid heat leaks from the heat exchanger, the tube wall should be thin and made of low conductivity material (e.g. stainless steel). For T 00F downstream of the heat exchanger, the situation is different due to the variable heat load Q_ . Here, the fluid mixing is important to enable measurement of the mean temperature. In the case of cryogenic fluids, the flow profile in the main stream is usually turbulent, which supports the mixing and hence requires a shorter distance between the heat exchanger and the downstream temperature sensor as compared to laminar flow. Both temperature sensors T 0F and T 00F are installed in copper blocks that enclose the tube completely in order to measure the circumferential average. 5.6. Specific heat capacity cp It was assumed earlier that the fluid’s specific heat capacity cp is known. As in any other thermal method, the fluid’s temperature change by adding heat depends on the specific heat capacity, thus cp cannot be eliminated.11 This restricts all thermal methods to measurements in single-phase flow. In order to determine the mass flow rate, the value of cp must be provided by fluid property data, which is straightforward in case of pure fluids. For mixtures, on the other hand, cp is composition-dependent. A theoretical option for measuring the true mass flow rate rather _ cp is the evaluation of time conthan the capacitance flow C_ ¼ m stants when ramping the heater power. In this case, test stand calibrations could be conducted with, e.g. helium and argon, featuring cp;He =cp;Ar  10=1 at 300 K and low pressure. The results could provide sensor-specific dependencies for the mass flow measurement of mixtures. The sensitivity of this option is strongly design-dependent and is yet to be investigated. 5.7. Temperature measurement The usual goal of temperature measurement is an accurate measurement of the object’s temperature. However, temperature sensors always measure their own temperature, and the sensor operation normally requires a trade-off between the parameters shown in Fig. 12. In resistance thermometers, low current yields low self-heating but poor resolution, and vice versa. In case of the method presented in this paper, accurate temperature measurement is actually not required. In fact, the values of systematic errors (offsets) do not matter at all, as they are fully compensated by the solution algorithm. What matters is the resolution of the temperature measurement system. Resistance thermometers can therefore be operated at the outer circumference of Fig. 12, yielding higher resolution due to larger signal-to-noise ratios compared to operating parameters closer to the center. With regard to resolution, temperature measurement with vapor pressure sensors would be the ultimate solution, achieving resolutions of DT  1 mK (see e.g. [5]). A corresponding design concept is presented in Fig. 13, where the constant surface temperature is achieved by the saturation temperature of the condensing fluid, equivalent to the principle in Fig. 8. With such a sensor design, the vapor pressure measurements of T 0F and T 00F exclude selfheating. Due to the wide operating temperature range from c. 300; . . . ; 4:2 K, however, resistance thermometers will have to be used in full-range cryogenic flow sensors. 5.8. Standard uncertainty Given resolution and noise levels of the temperature and heater power measurements influence the number of readings 11

Eq. (1) is essentially the definition function of cp .

Reproducibility Resolution

Statistical error

set Offematic

Readings

t Sys error

True value

Fig. 12. Common representation of parameters influencing the accuracy of temperature measurement with resistance thermometers.

Differential pressure gauge

Absolute pressure gauge p

Δp

Δp

Heat exchanger

Condensing film

Vapour pressure sensor

Saturated fluid

z

. Q

Fig. 13. Design principle based on vapor pressure measurements. The operating temperature range is restricted by the two-phase region of the chosen working fluid, and by the maximum working pressure.

ðsamplingrate  timeÞ and the minimum temperature differences required to reach a certain standard uncertainty for the mass flow rate. As the standard uncertainty only depends on random errors, the accuracy level can be influenced by statistics (i.e. the number of readings) in each measuring point (cf. Step A). Different accuracy levels can thus be obtained with the same sensor, defining e.g. ‘‘standard mode’’ and ‘‘precision mode’’ measurements. In theory, any accuracy level can be reached with this method, as long as sufficient statistics is provided. 6. Conclusions and outlook A new thermal method for mass flow measurement in pipes was presented. Its outstanding property is the capability for intrinsic calibration, based on analytic principles. This implies that calibrations can be executed at any moment and in any location, for instance in predetermined cycles during operation of a cryogenic system. With no need for a reference standard, the only prerequisite is stationary flow for some seconds during calibration, whereby this condition can be verified by the data. The resulting mass flow rate is free of systematics and the actual standard uncertainty of typically um_ < 1% is provided with the result. A proof-of-principle was presented, derived from experiments at 30 K in a largescale cryogenic system. The new method was filed for patent applications [8] by the Karlsruhe Institute of Technology (KIT). The capability for intrinsic calibration opens a new field of possibilities. It allows, for instance, to create a truly adaptive sensor, storing the results of each calibration in a database. These data provide an incremental mesh refinement of sampling points, through which empirical functions can be fitted for transient flow measurement. The accuracy can thus be increased or verified dur-

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S. Grohmann / Cryogenics 60 (2014) 9–18

ing operation of the system. This property distinguishes the new method fundamentally from other measuring principles, where the user has no possibility to verify the actual measurement uncertainty in his installation as compared to catalogue data, or to correct accuracy degradation by aging and contamination. The implementation of the new method is particularly interesting in cryogenic applications, where an external flow sensor calibration with low-temperature helium or hydrogen is rather unaffordable. KIT has started a technology transfer project together with the WEKA AG [9]. The project goal is to provide commercial cryogenic mass flow sensors within 2 years. References [1] Micro Motion Inc., Micro MotionÒ ELITEÒ Coriolis Flow and Density Meters, Product Data Sheet; 2013.

[2] Polifke W, Kopitz J, Wärmeübertragung – Grundlagen, analytische und numerische Methoden, 2nd ed. Pearson Studium; 2009. [3] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. 5th ed. John Wiley & Sons; 2002. [4] Grohmann S. Stability analyses of the beam tube cooling system in the KATRIN source cryostat. Cryogenics 2009;49:413–20. http://dx.doi.org/ 10.1016/j.cryogenics.2009.06.001. [5] Grohmann S, Bode T, Hötzel M, Schön H, Süßer M, Wahl T. The thermal behaviour of the tritium source in KATRIN. Cryogenics 2013;55–56:5–11. doi: http://dx.doi.org/10.1016/j.cryogenics.2013.01.001, http://dx.doi.org/http:// dx.doi.org/10.1016/j.cryogenics.2013.01.001. [6] Lemmon EW, Huber ML, McLinden MO. NIST standard reference database 23: reference fluid thermodynamic and transport properties – REFPROP, 9.1 ed. 2013. [7] Wolfram Research. MathematicaÒ, 9.0 ed. 2013. [8] Grohmann S. Vorrichtung und Verfahren zur Bestimmung des Massenstroms eines Fluids. German Patent Application DE 102011120899 A1, also published as WO 002013087174 A1; 06 2013. [9] WEKA AG, 8344 Baeretswil, Switzerland, .