A time-saving method to evaluate the thermal insulation performance of cryogenic vessels

Journal of Cleaner Production 258 (2020) 120632

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Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro

A time-saving method to evaluate the thermal insulation performance of cryogenic vessels J.F. Zhang a, b, G.M. Wei a, b, Z.G. Qu a, b, * a b

Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, China School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 November 2019 Received in revised form 7 January 2020 Accepted 16 February 2020 Available online 20 February 2020

Equivalent mass loss is a critical index used to evaluate the thermal insulation performance of cryogenic vessels. The current test standards for measuring the thermal insulation performance of cryogenic vessels in different countries require a long testing time and waste a considerable amount of working medium. In this study, a method that can shorten the test time of the thermal insulation performance of cryogenic vessels is established. Different from the current test methods that rely entirely on experiments, the method proposed in this paper is a semi-experimental and semi-analytical calculation method. The proposed method is based on the one-dimensional heat transfer analysis of a cryogenic vessel and short-term experimental test data, and it can be used to predict the mass loss over a long period of time. Moreover, the method can predict the mass loss of a cryogenic vessel at a high liquid level when the mass loss at a low liquid level is known. When this method is used to predict the mass loss, the comparison with the experimental data shows that the maximum relative error of the instantaneous mass flow is 21.1% and the relative error of the cumulative mass flow is less than 8%. When this method is used for the calculation of heat leakage between different liquid levels, the results show that the larger the difference between the two liquid levels, the greater the error of leakage heat, and the maximum is 18.9%. © 2020 Elsevier Ltd. All rights reserved.

Handling Editor: Weidong Li Keywords: Cryogenic vessel Prediction method Mass loss Heat leakage Thermal insulation performance Experimental test

1. Introduction Cryogenic vessels are specifically designed to store cryogenic liquids at a low temperature, such as
196  C for liquid nitrogen and
252  C for liquid hydrogen. Cryogenic vessels are widely used in different fields and applications, such as for storing the cryogenic liquid propellants of rocket engines; as a liquid hydrogen fuel tank for cars (Zohuri, 2019); in cryoelectronics; in cryobiology (Balasubramanian et al., 2012); and as part of the hydrogen storage system (Acar and Dincer, 2019). The huge temperature difference between the internal cryogenic liquid and the external environment causes an extremely strong heat transfer between the two, which must be prevented by the thermal insulation material to ensure secure storage of the cryogenic liquid (Peterson and Weisend, 2019). Without a good insulation structure, a large amount of storage medium (e.g., liquid hydrogen, liquid nitrogen)

* Corresponding author. Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, China. E-mail address: [email protected] (Z.G. Qu). https://doi.org/10.1016/j.jclepro.2020.120632 0959-6526/© 2020 Elsevier Ltd. All rights reserved.

will inevitably escape from the cryogenic tank (Moreno-Blanco et al., 2018; Heidary Moghadam et al., 2015). Reducing the heat leakage from cryogenic vessels has been studied by focusing on the insulation materials and structure of the vessels, and establishing the corresponding heat leakage calculation models. For example, Wang et al. investigated the thermal insulation performance of four different density Hollow glass microspheres (HGMs) (Wang et al., 2019); Wang et al. (2016) investigated the thermal insulation properties of variable density multilayer thermal insulation (VDMLI) materials with different configurations and spacers; Zheng et al. (2019) introduced selfevaporating vapor cooling shields (VCS) in spray on foam insulation (SOFI), multilayer insulation (MLI), and variable density MLI (VDMLI) systems to recover their sensible heat, thereby improving the thermal insulation performance; Johnson (2010) and Hastings et al. (2004) established heat transfer models for multilayer insulation systems. Yury Alexandrovich Kostikov et al. (Kostikov and Romanenkov, 2019) presented an optimization scheme for onedimensional parabolic equations with mixed boundary conditions for the control problem of uniform sphere heat conduction. These

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J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632

Nomenclature A Ac cg cv D G Gr h H k m mout m_ out Nu p Pr qv Q Q_

Convective heat transfer area, m2 Sectional area, m2 Velocity of gas at the outlet, m/s Isochoric heat capacity, J/(kg∙K) Internal diameter, m Acceleration of gravity, 9.8 m/s2 Grashof number Convective heat transfer coefficient, W/(m2∙K) Specific enthalpy, J/kg Heat transfer coefficient, W/(m2∙K) Mass of working medium, kg Total mass loss, kg Mass flow rate of product, kg/s Nusselt number Pressure, kPa Prandtl number Gaseous volumetric discharge rate, m3/s Total heat leakage, J

r RGLA T u U v

Latent heat of vaporization, J/kg Ratio of gas to liquid heat leakage Temperature, K Specific internal energy, J/kg Total internal energy of the system, J Viscosity, m2/s

Greek Symbols av Thermal expansivity, 1/K d Error, % db Boundary layer thickness, m l Thermal conductivity, W/(m∙K) r Density, kg/m3 t Time, s Subscripts g l s w

Gas Liquid Saturated state Wall

Instantaneous heat leakage, W

models are generally used for the design calculation of cryogenic vessels and are mainly applied to cases where the vessel is fully filled with liquid. In the process of thermal insulation performance testing, the cryogenic vessels are not necessarily fully filled with liquid. For cases considering gas-phase heat transfer, finite element analysis is generally used (Seo and Jeong, 2010; Wang et al., 2015), which has high requirements on computing resources and is relatively complicated. Therefore, the above models are not convenient for engineering applications. In addition to using the heat leakage to judge the thermal insulation performance of cryogenic vessels, the mass loss is also specified as a criterion in the test standards in different counties (BS EN 13530-2-2002; ISO 21014-2006; GBT, 18443.5-2010). However, in these standards, it is stipulated that the cryogenic vessel should be completely filled and left standing for at least 48 h before the test, and then the mass loss should be recorded for at least 24 h. In addition, the remaining working medium is discharged and cannot be recycled when the test process is completed. This process consumes a considerable amount of time and working medium. Based on the above information, the current model or test method for the thermal insulation performance of cryogenic vessels is complicated, time consuming, and a waste of working medium. The purpose of this paper is to introduce a simplified method to predict the thermal insulation performance of cryogenic vessels, and ultimately achieve the goal of saving time and working medium during the test process. In section 2, the model establishment and method implementation steps are presented. The accuracy of the optimized test method for insulation performance of cryogenic vessel is shown in section 3. The conclusions are drawn in section 4.

2.1. One-dimensional heat transfer model In general, cryogenic vessels have an inner shell and an outer shell, each with two dished ends and a cylindrical section in the middle, as shown in Fig. 1. The gap between the two shells is also known as a vacuum jacket (Seeli et al., 2016). The air in the gap is sucked out by a vacuum pump for better insulation. The vacuum jacket can be filled with different insulation materials and arranged in different forms, such as HGMs (Wang et al., 2019), VDMLI (Wang et al., 2016) and so on. Compared to the heat storage thickness of a
lu, 2019) reinforced concrete wall of 20 cm thickness (Topçuog which frequently, such double-shell vacuum structure greatly reduces the volume of the container and to obtain a better insulation effect. When calculating its heat flowrate across the shell of

2. Prediction model The prediction model is based on a one-dimensional heat transfer analysis. Furthermore, a dimensionless quantity describing the ratio of the gas-phase heat leakage to the liquid-phase heat leakage in the cryogenic vessel is also proposed.

(a)

(b)

Fig. 1. Structure of vertical cryogenic vessel: a) Diagram of vertical cryogenic vessel, b) Picture of cryogenic vessel.

J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632

cryogenic vessel, the heat transfer process can usually be regarded as one-dimensional heat transfer (Chen et al., 2004). To calculate the heat leakage and mass loss, the following hypothesis is made for the heat transfer model of a cryogenic vessel: (1) The gas and liquid phases are in equilibrium; (2) The system pressure is the liquid saturation pressure, which is consistent with the ambient pressure; (3) Both the gas and liquid temperature are the saturation temperature. Based on the above assumptions, the heat leakage from the outside to the inside of the cryogenic vessel can be described by a one-dimensional model. Taking a cryogenic vessel as an example, the calculation model is shown in Fig. 2, where dQ represents the total heat leakage over a time period of dt; Hgdmout represents the heat absorbed by the vaporization of the liquid; and dmout denotes the mass of the gas escaping over a time period of dt. The total energy Q gained inside the vessel over a period of time is:

Q ¼ m_ out ,r,t;

(1)

where Q represents the total heat leakage within a certain time period, m_ out is the instantaneous mass flow rate of the product by evaporation, r represents the latent heat of vaporization, and t is the given time period. Equation (1) shows the relationship between the mass loss by evaporation and the total heat leakage in a given time period. The energy conservation equation, system internal energy change, and mass conservation equations are as follows (Cengel and Boles, 2015):

dU dQ dmout 1 dmout 2 ¼
Hg
c ; dt dt 2 dt g dt

(2)

 dU d
¼ mg ug þ ml ul ; dt dt

(3)





dmg dml dmout ; ¼
þ dt dt dt

(4)

where U is the rate of change in the system energy, Hg is the specific

3

enthalpy of the gas, mout is the mass of the gas escaping by evaporation, cg is the velocity at the outlet of the vessel, and u is the specific energy of the working medium. Equations (2)e(4) can be solved through the following combination:

dmout 1 dQ  ¼  dt dt 1 2 a r þ 2cg 0 1 bugl m ml dTs g  @ cv:g þ cv:l þ  A þ ð
1Þ  ; m dt r l 1 a r þ 2c2g ml r l
1

(5)

g

where a ¼

1 , r 1
rg l

  2  dr mg rl drl l b¼m , dTg þ dT , r ¼ Hg
Hl , ugl ¼ r , ml , r l

g

s

s

ug
ul . Considering that the mass of the liquid in the cryogenic vessel is much larger than the mass of the gas, then ml > > mg . As the system pressure is approximately equal to the atmospheric pressure, which can be considered to remain basically unchanged, bugl  < < cv:l . Therefore, Eq. (5) can be simplified to the

then ml

rl rg
1

following:

1 mc dT  Q_ þ ð
1Þ  l v:l , s ; m_ out ¼  dt a r þ 12c2g a r þ 12c2g

(6)

where Q_ represents the instantaneous heat leakage from the cryogenic vessel. The saturation temperature, Ts, in Eq. (6) was obtained from the NIST Reference Fluid Properties database (REFPROP 9.1). Considering that the liquid level of the cryogenic vessel will change in the evaporation process, the liquid mass ml in the cryogenic vessel was calculated according to the following Eq. (7).

ml ¼ ml:0
m_ out ,t;

(7)

where ml:0 means the mass of the liquid in the vessel at the beginning of the test. To determine the Q_ in Eq. (6), a ratio of dimensionless quantities is proposed, which is detailed in the next section.

2.2. Heat leakage ratio of gas and liquid phases (RGLA) In Eq. (6), the instantaneous heat leakage Q_ consists of two parts, namely, the gas-phase heat leakage and the liquid-phase heat leakage. Therefore, the total heat leakage can be calculated by Eq. (8) as follows:

Q_ ¼ Q_ g þ Q_ l :

(8)

In the formula, the subscripts g and l represent the gas phase and the liquid phase, respectively. The liquid-phase heat leakage Q_ l can be calculated by Eq. (9) as follows:

Q_ l ¼ kl Al ðTamb
Ts Þ;

Fig. 2. Simplified model for heat and mass transfer process in cryogenic vessel.

(9)

where kl , Al , Tamb , and Ts are the total heat transfer coefficient of the liquid phase, liquid-phase heat transfer area, ambient temperature, and saturation temperature, respectively. The heat leakage of the gas phase is difficult to calculate owing to the temperature stratification in the gas-phase space. Therefore,

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J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632

this study introduces a ratio coefficient (RGLA) to solve the problem as shown in the following Eq. (10):

    Q_ g Nug lg DTg l Ag , l, : RGLA ¼ ¼ Nul ll DTl lg A l Q_

(10)

l

The dimensionless number RGLA shows the ratio of the heat leakage in the gas to liquid phases. The first term on the right of the equation represents the heat transfer performance of the gas/liquid phases, and the second term represents the influence of the liquid level change on the heat leakage of the gas/liquid phases. lg and ll represent the qualitative dimensions of the gas phase and the liquid phase, respectively; DTg represents the average temperature difference between the gas phase and the inner wall; DTl represents the average temperature difference between the liquid phase and the inner wall; and Ag and Al represent the contact areas of the gas phase and the liquid phase with the inner wall, respectively. At this point, Eq. (8) can be rewritten as Eq. (11):

Q_ ¼ Q_ l ,ð1 þ RGLA Þ:

2.3. Prediction method procedure Based on the above model and dimensionless quantity, we propose a prediction method. Taking short-term experimental data as samples, the method can be used to predict the mass loss of the working medium at a specific liquid level over a long period of time. In addition, if the mass loss of a cryogenic vessel in a low liquid level state is known, the method can be used to predict the mass loss of this cryogenic vessel when it is in a high liquid level state. 2.3.1. Prediction method for the mass loss at a specific liquid level This section describes a method for reducing the test time at a specific liquid level. According to the relevant formulas in Section 2.1 and Section 2.2, the following steps are constructed for the purpose of rapid prediction. A flowchart can be seen in Fig. 3. A. Fill the cryogenic vessel, leave it standing for 48 h, and record the ambient temperature and ambient pressure simultaneously; B. Record the gaseous volumetric discharge rate qv of the vessel for 6 h, and record the ambient temperature Tamb , ambient pressure

(11)

The contact area of the gas phase and the inner wall, Ag , in Eq. (10) can be calculated according to the integral formula provided by Wang and Wang (2010). The same method is used for the contact area, Al , between the liquid phase and the inner wall. The liquid temperature is considered the saturation temperature. The temperature of the gas phase in the cryogenic vessel can be directly measured using a thermocouple. The thermal conductivity (lg and ll ) of the gas phase and liquid phase can be obtained from the NIST Reference Fluid Properties database (REFPROP 9.1). The next step is to solve the Nu in Eq. (10). According to Yu et al. (1992) and Bostock and Scurlock (2019), when the internal diameter D and the boundary layer db of horizontal vessels meet in Eq. (12), the heat exchange between the working medium and the inner wall can be described by natural convection in an infinite space. It is found that this rule also applies to vertical vessels (Choi et al., 2016).

2db < 0:02 D

(12)

Therefore, the heat transfer mode between the gas/liquid phases inside the cryogenic vessel and the inner wall is natural convection. The Nu number and Gr number can now be calculated according to the following Eqs. (13) and (14) (Yang and Zhang, 1994):

Nu ¼ CðGr,PrÞn

Gr ¼

(13)

g av DTl3 v2

(14)

where av , DT , l, and v are thermal expansivity of fluid, temperature difference between fluid and wall, qualitative dimensions and viscosity, respectively. The values of parameters C and n in Eq. (13) are determined for Gr, as shown in Table 1 (Yang and Zhang, 1994).

Table 1 The constants C and n in Eq. (13). C

n

Scope of application of Gr

0.59 0.0292 0.11

1/4 0.39 1/3

1.43  104e3.9  109 3  109e2  1010 >2  1010

Fig. 3. A flowchart for reducing the test time at a specific liquid level.

J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632

5

Pamb , gas space temperature Tg , and initial liquid level simultaneously; The mass flow m_ out is calculated as follows:

m_ out ¼ qv ,rg

(15)

It should be noted that m_ out in Eq. (15) is measured experimentally and used to calculate the leakage heat Q_ for the first 6 h. C. Calculate the average heat leakage Q_ and the RGLA number, respectively, based on the recorded data in step B according to Eq. (6) and Eq. (11); D. Calculate the heat transfer coefficient kl of the liquid phase according to Eqs. (9) and (11), and average the kl ; The average heat transfer coefficient of the liquid phase kl is used to calculate the gaseous mass loss rate m_ out for the remaining time by Eqs. (6) and (11). The parameters such as ambient temperature and pressure referred to in Eq. (6) can utilize the values from the same time on the previous day recorded in Step A. The integral calculation of Eq. (6) can obtain the total mass loss mout . Compared with the existing test methods, the proposed method reduces the detection time from 24 h to 6 h (6 h is determined by previous comparative studies). It should be noted that the 48 h in Step A is to stabilize the working medium in the cryogenic vessel after filling, which could not be shortened. 2.3.2. Prediction of mass loss at high liquid level The heat leakage from the cryogenic vessel is related to the liquid level. After the change in liquid level, the heat leakage of the gas phase and liquid phase will also change. The existing test methods are based on a high liquid level and the working medium is discharged after the test. Therefore, if the mass loss at a high liquid level can be predicted based on the data from a low liquid level, a significant amount of working medium can be saved. Through the above analysis, the heat leakage in the gas phase and liquid phase can be calculated from the convective heat transfer equation. Thus, Eq. (8) can be rewritten as Eq. (16):


 Q_ ¼ Q_ g þ Q_ l ¼ hg Ag Tw:g
Tg þ hl Al ðTw:l
Tl Þ:

(16)

Based on the dimensionless quantity RGLA proposed above, the heat leakage from the cryogenic vessel at different liquid levels can be quickly calculated through the RGLA, and the equation is as follows:


 Q_ 1 Q_ 1;g þ Q_ 1;l ð1 þ RGLA1 ÞQ_ 1;l ð1 þ RGLA1 Þ h1:l A1;l DT1;l
: ¼ ¼ ¼ Q_ 2 Q_ 2;g þ Q_ 2;l ð1 þ RGLA2 ÞQ_ 2;l ð1 þ RGLA2 Þ h2:l A2;l DT2;l (17) Subscripts 1 and 2 represent two different liquid level states. Because the liquid in the cryogenic vessel is saturated, the temperature difference between the liquid and the wall and the convective heat transfer coefficient can be considered to be substantially constant. Therefore, Eq. (17) can be simplified to the following:

Q_ 1 ð1 þ RGLA:1 Þ A1:l ¼ : Q_ 2 ð1 þ RGLA :2 Þ A2:l

(18)

Assuming the acquisition of experimental data at a low liquid level and the calculation of the heat leakage at the low liquid level

Fig. 4. Flowchart to shorten test time based on low liquid level.

by the above-described steps, it is possible to further calculate the heat leakage at the high liquid level by Eq. (18). In this way, the goal of saving time and working medium can be achieved. The brief implementation procedure is as follows: Follow steps AeD in Section 2.3.1; the parameters Q_ and kl in the first 6 h at a low liquid level is obtained. Then, calculate the mass loss at the high liquid level as follows: E. The average heat transfer coefficient of the liquid phase kl is used to calculate the heat leakage Q_ for the remaining time using Eqs. (9) and (11) at a low liquid level. F. The heat leakage Q_ at a high liquid level within 24 h is calculated using Eq. (18). G. The gaseous mass loss rate m_ out for the remaining time is calculated using Eq. (6). The integral calculation of Eq. (6) can obtain the total mass loss mout at a high liquid level. Its flowchart can be seen in Fig. 4. 3. Validation of prediction method In order to verify the accuracy of the proposed method, this paper verified the method using the data in Yang et al. (2010) and the data provided by the Guangdong Institute of Special Equipment

Table 2 Parameters of cryogenic vessel. Nominal capacity (L)

Working pressure (Mpa)

Materials

Internal diameter (mm)

Internal length (mm)

Head height (mm)

175

1.38

SUS 304

455

915

129.4

6

J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632 -5

-5

5.0x10

5.0x10

Experiment data from reference (Yang et al., 2010) Calculated result by proposed method

-5

4.5x10

-5

-5

4.0x10

Mass flow rate(kg/s)

4.0x10

Mass flow rate(kg/s)

Experiment data from reference (Yang et al., 2010) Calculated result by proposed method

-5

4.5x10

-5

3.5x10

-5

3.0x10

-5

2.5x10

-5

2.0x10

=18.7%

-5

3.5x10

-5

3.0x10

-5

2.5x10

-5

2.0x10

=10.0%

max

max -5

-5

1.5x10

1.5x10

-5

1.0x10

-5

0

5

10

15

1.0x10

20

0

5

Time(H)

10

15

20

Time(H)

(a)

(b) -5

5.0x10

Experiment data from reference (Yang et al., 2010) Calculated result by proposed method

-5

4.5x10

-5

Mass flow rate(kg/s)

4.0x10

-5

3.5x10

-5

3.0x10

-5

2.5x10

=14.0%

-5

2.0x10

max

-5

1.5x10

-5

1.0x10

0

5

10

15

20

Time (H)

(c) Fig. 5. Comparison of the instantaneous mass flow rate calculated by the proposed method with the literature data at a specific liquid level under different ambient temperature conditions: (a) 10  C, (b) 21  C, (c) 30  C.

1.80

Experiment data from reference (Yang et al., 2010) Calculated result by proposed method

1.75

Total loss of mass in 18h (kg)

1.70 1.65 1.60

Inspection and Research Dongguan Branch. In the data provided by the Dongguan Branch, the volume of the cryogenic vessel used in the experiment is 175 L. The detailed parameters of the cryogenic vessel are shown in Table 2. The working medium was liquid nitrogen. The entire experiment was carried out in a thermostat chamber. The experiment selected five temperatures: 10  C, 16  C, 25  C, 30  C, and 35  C, as different thermostat temperatures.

1.55

3.1. Validation of prediction method at a specified liquid level

1.50

=2.0%

max

1.45 1.40 1.35 1.30

10

20

30

Ambient temperature ( ) Fig. 6. Comparison of total mass loss between the calculated data and experimental data in reference at different ambient temperatures.

The proposed method was first validated by using the experimental data in Yang et al. (2010). The working medium used in the reference is liquid nitrogen. All the parameters required for the calculations in the prediction method are based on the reference. The experimental data of the first 6 h in the reference were selected as the sample data, which were used to predict the mass loss for the next 18 h. The predicted results are compared with the experimental data from the reference in Fig. 5. Fig. 5 shows that the lower the ambient temperature is, the greater the maximum error in the instantaneous evaporation flow will be, and the higher the ambient

J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632 -5

7

-5

5.0x10

5.0x10

Experiment Prediction model

-5

4.5x10

-5

Experiment Prediction model

-5

4.5x10

4.0x10

-5

4.0x10

Mass flow rate (kg/s)

Mass flow rate (kg/s)

-5

3.5x10

-5

3.0x10

-5

2.5x10

=15.7%

max

-5

2.0x10

-5

1.5x10

-5

3.5x10

-5

3.0x10

=21.1%

max

-5

2.5x10

-5

1.0x10

-5

2.0x10

-6

5.0x10

-5

0.0 0

5

10

15

1.5x10

20

0

5

10

Time (H)

(a) -5

-5

5.0x10

Experiment Prediction model

-5

4.5x10

Experiment Prediction model

-5

4.5x10

-5

-5

4.0x10

4.0x10

Mass flow rate (kg/s)

Mass flow rate (kg/s)

20

(b)

5.0x10

-5

3.5x10

=9.0%

-5

3.0x10

max

-5

2.5x10

-5

3.5x10

-5

3.0x10

=10.6%

max -5

2.5x10

-5

-5

2.0x10

2.0x10

-5

-5

1.5x10

15

Time (H)

0

5

10

15

1.5x10

20

0

5

10

15

20

Time (H)

Time (H)

(c)

(d) -5

5.0x10

Experiment Prediction model

-5

4.5x10

-5

Mass flow rate (kg/s)

4.0x10

-5

3.5x10

-5

3.0x10

=11.6%

max -5

2.5x10

-5

2.0x10

-5

1.5x10

0

5

10

15

20

Time (H)

(e) Fig. 7. Comparison of the mass flow rate calculated by the proposed method with the experimental data provided by the Dongguan Branch at a specific liquid level under different ambient temperature conditions: (a) 10  C; (b) 16  C; (c) 25  C; (d) 30  C; (e) 35  C.

temperature is, the more accurate the model will be. The reason for this may be that when the ambient temperature is large, the stronger the heat transfer power is inside and outside the cryogenic

vessel, the temperature effect should be more and more obvious at this time, so the model agrees well with the literature value. In the figure, the instantaneous mass flow is taken as the ordinate and the

8

J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632

2.6

Experiment Prediction model

Mass flow (kg/d)

2.4

2.2

2.0

=7.1%

max

1.8

1.6

10

20

30

40

pressure is the decisive cause of mass loss fluctuations. It can be seen that the maximum relative error of the prediction method in Fig. 7 is 21.1% and the maximum average error is 8.6% under different temperature conditions. Fig. 8 indicates the total mass loss by evaporation in 18 h. The error of the total mass loss between the calculated data and the experimental data is also less than 8%. When the temperature of the thermostat chamber is above 25  C, the error is below 5%. Fig. 8 shows that the higher the external temperature of the cryogenic vessels, the larger the mass loss, and the influence of the temperature at this time is increased. The accuracy of the model is higher also. This conclusion is consistent with the conclusions compared with literature data. Therefore, it can be concluded that the above prediction method is reliable for the prediction of the mass loss by evaporation at a specific liquid level. Furthermore, the method can reduce the test time for evaluating the thermal insulation performance of cryogenic vessels.

Ambient temperature ( ) Fig. 8. Comparison of total mass loss between the calculated data and experimental data provided by Dongguan Branch at different thermostat temperatures.

1.2 1.1

Experiment results from reference Yang et al., 2010) Calculated results by proposed method

1.0

=6.0%

Q/Qr

0.9 0.8

=13.6% =18.9%

0.7 0.6 0.5 0.4

0.5

0.6

0.7

0.8

0.9

1.0

Liquid level Fig. 9. Relative error of the heat leakage between calculated data and literature data at different liquid levels.

time is taken as the abscissa. The maximum relative error of the prediction method in Fig. 5 is 18.7%, the maximum average error is 7.6% under different temperature conditions, and the average error is less than 10%. Fig. 6 indicates the total mass loss by evaporation in 18 h at different ambient temperatures. The error of the total mass loss between the calculated data and the experimental data is less than 5%. In addition to the comparison with the values in the reference (Yang et al., 2010), an experimental data provided by Dongguan Branch was also used to verify the prediction method. The results are shown in Figs. 7 and 8. Fig. 7 shows that as the ambient pressure fluctuates throughout the day, the rate of mass loss fluctuates, even when placed in a thermostat chamber. It means that ambient

3.2. Validation of prediction method for mass loss at different liquid level The accuracy of the proposed method for predicting the mass loss at a high liquid level based on the data from a low liquid level was validated by first comparing the results of the proposed method with the experimental data in Yang et al. (2010). In the reference study, four groups of tests were conducted for the mass loss by evaporation in a cryogenic vessel. The liquid levels of the four groups of experiments were 89%, 76%, 62%, and 48%, respectively. The proposed prediction method can use the data obtained at a low liquid level to calculate the mass loss by evaporation at a high liquid level, and vice versa. To facilitate the display of the comparison results, we used the heat leakage at the highest liquid level to predict the heat leakage at a lower liquid level. In Fig. 9, the heat leakage of the highest liquid level (89%) is selected as the reference value. The heat leakage at three lower liquid levels (76%, 62%, and 48%) is calculated using the data at the highest liquid level according to the proposed method. To better present the comparison between the calculated value and the experimental value obtained by Yang et al. (2010), we consider Q/Qr as the ordinate and the liquid level as the abscissa. Q is the heat leakage at a low liquid level and Qr is the heat leakage at the highest liquid level. It is seen that the maximum error of the heat leakage is 18.9%, and the error increases when the difference between the two liquid levels (reference liquid level and level to be solved) increases. Experimental verification was also carried out for a vertical vessel. In the experimental data provided by Dongguan Branch, three liquid levels of 27.2%, 31.5%, and 48.5% were selected as low liquid levels, and the proposed prediction method was used to predict the mass loss at a liquid level of 84.4% based on the data from the three low liquid levels. Table 3 displays the comparison of the mass loss calculated by the proposed prediction method with the experimental value in 24 h. It can be seen that all the calculated values are higher than the experimental values. This is because when the liquid level of the cryogenic insulation vessel is too low, the temperature difference between the gas and the inner wall increases, and the calculated gas-phase heat leakage may be larger than the actual value, and

Table 3 Mass loss by evaporation in 24 h at high liquid level calculated from parameters at three low liquid levels.

Mass loss (kg) Error (%)

Calculated by 27.2%

Calculated by 31.5%

Calculated by 48.5%

84.4% (Reference)

3.58 20.5

3.43 15.7

3.36 13.3

2.96 e

J.F. Zhang et al. / Journal of Cleaner Production 258 (2020) 120632

finally the mass loss error increases. However, within the scope of the present study, it seems that if the appropriately low filling level range is chosen (e.g., above 30%), the calculation error will be less than 20%, which is still acceptable for engineering applications. Most of all, based on the proposed method, if a lower liquid filling level (which means less product) could be used to evaluate the thermal insulation performance of a cryogenic vessel at a higher liquid filling level in the test process, a significant amount of working medium could be saved. 4. Conclusions The conventional method for testing the thermal insulation performance of a cryogenic vessel is performed entirely through experiments, so it takes a long time. Based on the one-dimensional heat transfer, a calculation model of the mass loss of the cryogenic vessel is established in this paper, and a set of procedures for shortening the test time of the cryogenic vessel is designed. The natural convection is used to derive the proportional relationship between the liquid level of the cryogenic vessel and the heat leakage. Combining this proportional relationship with a onedimensional calculation model, this paper proposes a method for shortening the test time of the thermal insulation performance based on a low liquid level. The proposed prediction method can be used to evaluate the thermal insulation performance of cryogenic vessels at a specific liquid filling level, and the method can also use the data at a low liquid level to predict the mass loss at a high liquid level. Through experimental verification, the maximum error of the total mass loss between the calculated and experimental data at a specified liquid level is less than 8%. When using the experimental data at a low liquid level to predict the mass loss by evaporation at a high liquid level, it is recommended that the low liquid level be no less than 30% so that the error can be controlled to below 20%. According to the above analysis, the proposed prediction method can achieve the goal of saving time and working medium in the test process for evaluating the thermal insulation performance of cryogenic vessels. In this paper, the proposed method has been experimentally verified. The next step is to try to extend it to practical applications and further improve it accordingly. Declaration of competing interest The authors declare that there are no financial or other relationships that might lead to a conflict of interest of the present article. All authors have reviewed the final version of the manuscript and approved it for publication. CRediT authorship contribution statement J.F. Zhang: Conceptualization, Methodology, Formal analysis, Writing - review & editing. G.M. Wei: Formal analysis, Data curation, Writing - original draft. Z.G. Qu: Conceptualization, Writing review & editing, Supervision, Funding acquisition. Acknowledgments This work was supported by funding from the National Natural Science Foundation of China (No. 51576155), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51721004) and the 111 project (B16038). The authors also express their gratitude to the Guangdong Institute of Special Equipment Inspection and Research, Dongguan Branch, for providing the experimental data.

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