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A novel heat pump system using a multi-stage Knudsen compressor
International Journal of Heat and Mass Transfer 127 (2018) 84–91
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A novel heat pump system using a multi-stage Knudsen compressor K. Kugimoto a,b,⇑, Y. Hirota a, T. Yamauchi a, H. Yamaguchi b, T. Niimi b a b
Toyota Central R&D Labs., Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan Department of Micro-Nano Mechanical Science and Engineering, Nagoya University, Chikusa Furo-cho, Nagoya, Aichi 464-8603, Japan
a r t i c l e
i n f o
Article history: Received 9 March 2018 Received in revised form 14 June 2018 Accepted 14 June 2018
Keywords: Microflow Heat recovery High Knudsen number Thermal molecular pump Thermal transpiration Performance prediction
a b s t r a c t A novel heat pump system is proposed using a multi-stage Knudsen compressor in the cycle between the evaporator and condenser. The proposed Knudsen heat pump is driven by thermal energy, and it is able to utilize waste heat. There are no moving parts in a Knudsen compressor, leading to several advantages, including a lack of vibration and noise, and high durability. In this study, the configuration necessary to achieve a performance such that it can be used as a practical heat pump (output power 1 kW, temperature difference 6 K) is considered. The performance of this heat pump is predicted by a onedimensional analytical model coupled with a simple experimental result. This method is based on our previously constructed method to predict the performance of a multi-stage Knudsen compressor, and the evaporation and condensation and the time variations of temperature in the evaporator and condenser are newly considered. A heat pump with a 30-stage Knudsen compressor using glass-fiber filters with an area of 4.00 m2 is predicted to generate an output power of 1.27 kW and a temperature difference of 6.00 K. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction In a rarefied gas regime, a temperature gradient of a channel wall induces a flow, which is called a thermal transpiration [1]. Knudsen initially observed experimentally the phenomenon at low pressures [2,3], and many studies were performed for academic interest after the first report [4–10]. Recently, the utilization of microporous materials and microfabrication technology has enabled the generation of thermal transpiration at high pressures, such as at atmospheric pressure. This phenomenon is utilized to realize a pump/compressor, which is called a Knudsen pump/compressor. A Knudsen compressor is driven by thermal energy, and it is able to utilize waste heat or environmental heat energy as a driving source, so it is expected to be a new class of heat recovery device. In addition, it has various advantages, such as a lack of vibration and noise, high durability, and easy miniaturization compared to other existing compressors because it has no mechanically moving parts. Several practical applications of a Knudsen pump/compressor have been proposed [11–19]. For example, Liu et al. and Qin and Gianchandani investigated a micro-gas chromatography system ⇑ Corresponding author at: Toyota Central R&D Labs., Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan. E-mail addresses: [email protected] (K. Kugimoto), e1476@mosk. tytlabs.co.jp (Y. Hirota), [email protected] (T. Yamauchi), hiroki@ nagoya-u.jp (H. Yamaguchi), [email protected] (T. Niimi). https://doi.org/10.1016/j.ijheatmasstransfer.2018.06.072 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
with a miniature Knudsen pump [11,12]. Nakaye and Sugimoto demonstrated a gas separator composed of Knudsen pumps [15]. However, applications of Knudsen pumps/compressors are still limited. One engineering product that uses a gas compressor is a heat pump. A heat pump system performs cooling and heating using the latent heat of a refrigerant, and it is used for air-conditioners, refrigerators, and the like. In general, an electrical compressor is used to generate a pressure difference between the evaporator and condenser so as to circulate the refrigerant vapor and to promote evaporation and condensation of the refrigerant. For heat recovery purposes, a heat-driven gas compressor may be used. An adsorption heat pump (AHP) is one such system [20]. In this system, the refrigerant vapor is sucked from the evaporator by adsorption of the adsorbent, and it is discharged to the condenser by desorption. The adsorbent is cooled with cooling water during adsorption, and it is heated with a waste heat medium during desorption. Therefore, it is necessary for AHP to repeatedly switch the gas flow path and the temperature control liquid flow path of the adsorbent, so many switching mechanisms are required. It may be possible to create a heat pump system with a simpler mechanism for heat recovery purposes by using a Knudsen compressor which can generate a practical pressure difference and mass flow. This study proposes a novel heat pump system using a multistage Knudsen compressor with porous materials as a compressor in the heat pump cycle, with water vapor as a refrigerant. The
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
transport of water vapor by a compressor was verified by our previous measurements [21]. However, the throughput of the Knudsen compressor is known to be low compared with a conventional mechanical compressor in a meter-scale system. Therefore, we predict the performance of the proposed heat pump system and clarify the configuration necessary to achieve the practical output power and temperature difference of conventional systems (e.g., 1 kW, 6 K). This prediction is made using the method for the multi-stage Knudsen compressor proposed in our previous study [22], with some refinements. This method was based on the one-dimensional analytical model from the mass conservation law for a multi-stage Knudsen compressor combined with a simple experimental result for a single stage, aiming to simulate a realistic performance of a high flow rate Knudsen compressor using a porous material with complicated aggregate channels. In this study, the prediction method is modified by adding the evaporation/condensation and the temperature time variations of the evaporator/ condenser. The output power of the proposed heat pump depends on the mass flow rate generated by the Knudsen compressor, while the temperature difference depends on the pressure difference. Therefore, the performance is evaluated to clarify the area of the porous material and the number of stages necessary to achieve a practical performance of an output power 1 kW and a temperature difference 6 K. 2. Heat pump system using a multi-stage Knudsen compressor 2.1. Knudsen compressor The degree of rarefaction of gas is expressed by the Knudsen number Kn. The Knudsen number of a gas in a channel is defined by the ratio of the mean free path of gas l to the characteristic length of the channel D.
Kn ¼
l : D
ð1Þ
The gas whose Kn is larger than 0.1 is considered to be a rarefied gas. Under this condition, a one-way gas flow is induced by the temperature difference between the two ends of the channel from the cold to hot side, which is called the thermal transpiration. (a) Microchannel
Kn>0.1
(b)
Thick channel
Kn<<0.1
Microchannel Connection part
85
An illustrative single-stage Knudsen compressor is shown in Fig. 1a. A single-stage Knudsen compressor is composed of a microchannel having a diameter DP of the same order of magnitude or less compared with the mean free path of gas and a channel having a diameter DC sufficiently larger than the mean free path. When given a mountain-shaped temperature gradient with a high temperature TH and a low temperature TC as in the lower figure of Fig. 1a, thermal transpirations are induced in both of the microchannel and thick channel, but a pressure-gradient driven counterflow results in no net flow only in the thick channel. Thereby, the Knudsen compressor can generate a one-way flow and a pressure difference between the inlet and outlet of the compressor, although there is no temperature difference between the inlet and outlet. Then, a multi-stage Knudsen compressor can be built by connecting multiple single-stage Knudsen compressors in series as in Fig. 1b. The possible pressure difference increases as the number of stages increases, while the mass flow rate is determined by the performance of the single stage. 2.2. Novel heat pump system The configuration of our novel heat pump system is shown in Fig. 2. Two chambers, referred to hereinafter as the evaporator and condenser, are filled with liquid water and water vapor as a refrigerant. A heat exchanger is installed within each chamber in order to take the heat inside the system to the outside. The evaporator and condenser are connected to each other by two channels to form a cycle: one channel in a vapor phase consists of the multistage Knudsen compressor, and the other in a liquid phase has a valve. When the Knudsen compressor is operated, vapor is made to flow by means of the compressor from the evaporator to the condenser. In the evaporator, the pressure drop due to the flow induced by the compressor promotes water evaporation and heat absorption due to the latent heat of water. Then, the temperature in the evaporator TE decreases corresponding to the decrease in the pressure in the evaporator PE from the saturated vapor pressure curve of water in Fig. 3, which is a monotonically increasing function. In contrast, in the condenser, the pressure increase promotes vapor condensation and heat generation. Then, the increase in the pressure in the condenser PC induces the increase in the temperature in the condenser TC. When the evaporator water decreases, the valve at the channel in a liquid phase is opened and liquid water is supplied from the condenser. As a result, it is possible to operate continuously in a completely isolated state from the outside, and there is no concern that dust will clog microchannels inside the Knudsen compressor. From the operating mechanism, the performance of the heat pump system is determined by the performance of the Knudsen compressor being used. The temperature difference obtained in this heat pump system depends on the pressure difference generated by the Knudsen compressors. The pressure difference can be amplified by cascading compressor units [14]. On the other hand, _ the power output Q depends on the mass flow rate of vapor M. _ In terms of the latent heat of water, L, Q is calculated as Q = LM. 3. Performance prediction method 3.1. Outline
Fig. 1. (a) Illustrative single-stage Knudsen compressor. (b) Illustrative multi-stage Knudsen compressor. The pressure difference is amplified by combining multiple single-stage Knudsen compressors in series.
The performance is predicted by modifying the prediction method for the multi-stage Knudsen pump/compressor in our previous study [22], where the practical performance was well predicted by the 1D analytical model based on the mass conservation law using the single-stage experimental results. The
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Water vapor
Evaporator
Condenser Multi-stage Knudsen compressor
Water vapor
Heat exchanger
Heat exchanger
Liquid water
Valve
Saturated vapor pressure (Pa)
Fig. 2. Device configuration of the heat pump system using a multi-stage Knudsen compressor.
8000
3.2. 1D model
6000
3.2.1. Derivation of time evolution recurrence formula of pressure distribution A schematic for the performance prediction model of the heat pump system using an N-stage Knudsen compressor is shown in Fig. 4. The mass conservation law applies to N + 1 spaces, which consist of two chambers and N 1 connection parts of the N-stage Knudsen compressor. A temperature difference is imposed on each Knudsen compressor unit so that the left side of the unit is at low temperature TL and the right side is at high temperature TH. The ith compressor unit from the cold side (left) is called ‘‘compressor(i)”. The space on the left of compressor(i) is called ‘‘space(i 1)”, and the space on the right is called ‘‘space(i)”; namely, the evaporator and condenser correspond to space(0) and space(N), respectively. Let Ai and Vi be the cross-sectional area of the porous material in compressor(i) and the volume of space(i), respectively. The time t(n) at the nth step is expressed using the time step Dt as ðnÞ ðnÞ _ ðnÞ are the pressure and average temt(n) = nDt. Let p , T , and M
4000
PC
2000
PE
0 273
TC
TE
293
313
Temperature (K) Fig. 3. Saturated vapor pressure curve of water [23]. Examples of PE, PC, TE, and TC are shown (steady-state values in calculations performed in Section 4.2).
water vapor is transported by the multi-stage Knudsen compressor, and evaporation in the evaporator and condensation in the condenser are newly considered in this study. The time histories of the pressure distribution inside the heat pump and temperatures in the evaporator and condenser are simulated.
Space(i-1) ( pi(−n1) , Vi −1, TM )
i
(n) 0
p;i
Space(i) ( pi( n ), Vi , TM )
Compressor(1) (Porous area: A1 , Compressor(i) Compressor(N) ( n) (AN , M p( n, N) ) Mass flow rate: M p( n,1) ) (Ai , M p ,i )
Space(0) Pressure: p
i
perature in space(i) at time t(n) and the mass flow rate generated by
Space(N)
p N(n ) VN
Volume: V0
TN(n )
(n)
Temperature: T0
(n)
Evaporation rate: M p , 0 Heat capacity: C0
Condensation rate:M p( n, N) +1
TL TH
CN
Fig. 4. Schematic for the performance prediction model of the heat pump system using an N-stage Knudsen compressor.
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K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91 ðnÞ
ðnÞ
compressor(i), respectively. T 0 and T N correspond to TE and TC, respectively. As for average temperature, it is simply referred to as ‘‘temperature” below. Different from the mass flow rate in other _ ðnÞ are mass flows generated by evaporation of _ ðnÞ and M spaces, M
3.2.2. Relational expression of pressure, temperature, evaporation and condensation rate in space(0) and space(N) There are three unknown values in space(0) and space(N), the pressure, temperature, and evaporation (or condensation) amount.
water in the evaporator and condensation of water vapor in the condenser, respectively. The temperature in the space(1), . . . ,
Equation (7) shows the relationship between the pressure p0
p;0
p;Nþ1
space(N 1) is assumed to be constant and ðnÞ
ðnÞ Ti
= TM = (TL + TH)/2
ðnÞ
(i = 1, 2, . . . , N 1); however, T 0 and T N are variables in this study. Heat capacities in space(0) and space(N) are denoted by C0 and CN. It is assumed that the heat capacity of the container is sufficiently larger than the heat capacity of water, and C0 and CN are fixed with respect to time. Using the mass of the gas in space(i) at the nth step and its timeðnÞ
ðnÞ
ðnþ1Þ
varying rate be Mc;i and dMc;i =dt, respectively, the mass M c;i th
ðnþ1Þ
M c;i
ðnÞ
ðnÞ
¼ M c;i þ Dt
dM c;i dt
ði ¼ 0; 1; . . . ; NÞ:
ð2Þ
The mass M c;i of the gas is obtained from the density qi ðnÞ
ðnÞ
the volume Vi, and q
ðnÞ i
ðnÞ
M c;i ¼ qi V i ¼ ðnÞ
ðnÞ pi ðnÞ RT i
Vi
and
is expressed as follows using the gas conðnÞ
and pressure pi :
stant of water vapor R, temperature T i ðnÞ
at
step can be expressed by the following equation:
the n + 1
ð3Þ
_ ðnÞ from space(i): space(i) and the outflow rate M p;iþ1
ði ¼ 0; 1; . . . ; NÞ:
ð4Þ
_ ðnÞ equation using mass flow rate per unit area m p;i and Ai:
Here,
ðnÞ _ p;i m
ði ¼ 1; 2; . . . ; NÞ:
ð5Þ
ðnÞ
ðnÞ
ðnÞ
difference across compressor(i) Dpp;i ¼ pi1 þ pi , determined from the experimental results of a single-stage Knudsen compressor, which is described later, in Section 3.4.1. ðnÞ _ ðnÞ ðnÞ _ ðnÞ ði ¼ 1; 2; . . . ; NÞ: m p;i ¼ mp;i ðpp;i ; Dpp;i Þ
ð6Þ
By substituting Eqs. (3) and (4) into Eq. (2), the pressure at the n + 1th step can be calculated as follows: ðnþ1Þ
ðnþ1Þ
¼
¼
_ ðnÞ LM p;i
Ti
ðnÞ
Ti
ðnÞ
pi þ Dt
ðnþ1Þ
RT i Vi
ð7Þ However, Eq. (7) is not closed, because the evaporation and conden_ ðnÞ , respectively, and the _ ðnÞ and M sation amount per unit time, M p;0
ðnþ1Þ
ðnþ1Þ
p;Nþ1
and T N are unknown. In order to close this temperatures T 0 equation, these relational expressions are further needed. The method of determining these values is described in the next section, Section 3.2.2.
ð8Þ
ðnÞ
= Q N in the steady state, but these values may differ tranðnþ1Þ
siently. The temperatures T 0
ðnþ1Þ
and T N
at the n + 1th step are ðnÞ
ðnÞ
expressed by the following equation using Q 0 and Q N . ðnþ1Þ
¼ T 0 Dt
ðnþ1Þ
¼ T N þ Dt
TN
ðnÞ
ðnÞ
ðnÞ
Q0 ; C0
ð9Þ
ðnÞ
QN : CN
ð10Þ
Substituting Eq. (8) into Eqs. (9) and (10), the relational expressions of the temperature, evaporation, and condensation amount are obtained by the following equations. ðnþ1Þ
T0
ðnþ1Þ
TN
_ LM p;0 ðnÞ
ðnÞ
¼ T 0 Dt
ðnÞ
¼ T N þ Dt
C0
;
ð11Þ
_ ðnÞ LM p;Nþ1 CN
:
ð12Þ
Also, the pressure and temperature in space(0) and space(N) are related by the saturated vapor pressure curve of water. According to Antoine’s equation [23], these relationships are as follows:
log10 ðjpi
ðnþ1Þ
Þ¼ a
b ðnþ1Þ
Ti
c
ði ¼ 0; NÞ:
ð13Þ
Here, j, a, b, and c are constants which depend on substances and units. In the case of water vapor, when using Pa as the unit of and K as the unit of temperature T i , j = 103 pressure pi 1 0 3 1 Pa , a = 7.17 10 , b = 1.72 10 K, and c = 4.06 10 K. By solving these simultaneous equations, the evaporation and condensation amounts per unit time at the nth step and the pressure and temperature at the n + 1th step can be derived in space(0) and space(N). ðnþ1Þ
_ ðnÞ M _ ðnÞ Þ ði ¼ 0; 1; . . . ; NÞ: ðM p;i p;iþ1
ðnÞ
p;Nþ1
ði ¼ 0; NÞ;
is a function of the average pressure between the inlet
ðnÞ ðnÞ ðnÞ p;i and outlet of compressor(i) p ¼ ðpi1 þ pi Þ=2 and the pressure
pi
ðnÞ Qi
T0
Assuming that the surface temperature of each porous material _ ðnÞ (i = 1, 2, . . . , N) can be obtained by the following is uniform, M p;i
ðnÞ _ ðnÞ ¼ m _ p;i M Ai p;i
ðnÞ
The heat absorption and generation per unit time, Q 0 and Q N , are expressed by the following equation with the latent heat of _ ðnÞ . _ ðnÞ and M water L using M
ðnÞ
th
p;i
_ ðnÞ _ ðnÞ M ¼M p;i p;iþ1
and temperature at the n + 1 step and the evaporation _ ðnÞ at the n step (or pðnþ1Þ , T ðnþ1Þ and M _ ðnÞ ). If there amount M N N p;0 p;Nþ1 are four additional independent equations that describe these relationships (two for each of i = 0, N), the numbers of unknowns and equations become equal, resulting in a solvable problem. Here, the relationship between the temperature and the evaporation or the condensation amount is derived by assuming that the evaporator and condenser are completely insulated, and the changes in temperature there are only due to the latent heat of the evaporation and condensation. In addition, the relationship between the pressure and temperature is derived by assuming that the evaporator and condenser are designed to be sufficiently large. Under this assumption, the pressures in space(0) and space(N) are always equal to the saturated vapor pressure of the temperatures there.
Q0
ðnÞ dM c;i =dt
at space(i) in the n _ ðnÞ to step is expressed by the difference between the inflow rate M
dt
th
p;0
ði ¼ 0; 1; . . . ; NÞ:
The time-varying rate of the mass
ðnÞ dM c;i
ðnþ1Þ
ðnþ1Þ T0 th
ðnþ1Þ
3.3. Numerical solution of simultaneous equations by Newton’s method ðnþ1Þ ðnþ1Þ _ ðnÞ (or pðnþ1Þ , T ðnþ1Þ , and M _ ðnÞ ) can be p0 , T 0 , and M N N p;0 p;Nþ1 obtained by simultaneously solving Eqs. (7) for i = 0, (11) and
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
(13) for i = 0 (or Eqs. (7) for i = N, (12) and (13) for i = N). However, algebraic analysis is difficult because Eqs. (7) and (13) are nonlin_ ðnÞ ) is numerically computed by Newton’s _ ðnÞ (or M ear. Thus, M p;0
p;Nþ1
method. _ ðnÞ is determined by the The analytic function f for solving M p;0 following equation using Eq. (7). ðnþ1Þ
ðnþ1Þ
_ ðnÞ ;pðnþ1Þ ;T ðnþ1Þ Þ ¼ pðnþ1Þ þ T 0 pðnÞ þ Dt RT 0 f ðM p;0 0 0 0 0 ðnÞ V0 T0
_ ðnÞ M _ ðnÞ Þ ¼ 0: ðM p;0 p;1 ð14Þ
ðnþ1Þ p0
_ ðnÞ M p;0
ðnþ1Þ T0
Here, and can be expressed as a function of from Eqs. (11) and (13). Thus, f can also be expressed as a function of _ ðnÞ . After that, the value of M _ ðnÞ is obtained by Newton’s method. M p;0
p;0
_ ðnÞ is determined, pðnþ1Þ and T ðnþ1Þ can be easily obtained Once M p;0 0 0 ðnþ1Þ ðnþ1Þ _ ðnÞ can be calculated from Eqs. (11) and (13). pN , T N , and M p;Nþ1 by the same process. 3.4. Relationship between mass flow rate and pressure difference of the single-stage Knudsen compressor 3.4.1. Performance of the Knudsen compressor The mass flow rate of the single-stage Knudsen compressor as a function of the average pressure between the inlet and outlet of the compressor and the pressure difference between them, as in Eq. (6), is derived as follows. Consider a system with a single-stage Knudsen compressor between two chambers with volume V. The time-dependent pressure difference between the chambers generated by the compressor is Dp, which is approximated by the following exponential function of t [24,25]:
t DpðtÞ ¼ DpMAX 1 es ;
ð15Þ
where DpMAX is the pressure difference at steady state, and s is a constant determined by comparing the experimental data and the _ can be derived fitting curve. The mass flow rate per unit area, m, in terms of the pressure difference between the two chambers by the constant volume method as follows [26]:
_ ¼ m
V d Dp ; RT R A dt 2
ð16Þ
where R is the gas constant of the gas in the compressor, TR is the room temperature, and A is the area of the porous material. The _ MAX , appears at t = 0 and Dp = 0 as maximum mass flow rate, m follows:
_ MAX ¼ m
V 1 Dp : 2RT R A MAX s
ð17Þ
_ be a By eliminating V, R, TR, and A from Eqs. (15)–(17), we let m _ MAX as follows: function of Dp with DpMAX and m
_ p ; DpÞ mð Dp ¼1 : Þ Þ _ MAX ðp DpMAX ðp m
material with micropores. The compressor unit is clamped by vacuum flanges with an insertion of square rings. Chilled water at TL or heated oil at TH circulates through the cooler or heater to keep the temperature constant. There are slits, which are the channels for gas, across the cooler and heater, so that the temperature of gas equilibrates to that of the cooler or heater. A glass-fiber filter used as the porous material is glued onto the cooler. The average pore size, thickness, and area of the glass fiber filter are 0.7 lm, 380 lm, and 4.90 cm2, respectively. To measure the performance of the single-stage Knudsen compressor, the single-stage Knudsen compressor is connected between two chambers with volumes of 5400 cm3. A closed loop is formed by connecting the two chambers with the Knudsen compressor and another pipe with a valve. The pressure of each chamber is measured by capacitance manometers (Setra 730G100T (F.S.: 13.3 kPa), 730G-010T (F.S.: 1.33 kPa), accuracy: ±0.150% of F.S.), which remain always connected to the chambers. The performance of the single-stage Knudsen compressor is evaluated by the constant volume method, which measures the time history of the pressure difference between these chambers as generated by the compressor. By closing a valve inserted into a pipe, the time-dependent pressure difference between the chambers, . Eq. (15), is recorded for a certain value of the average pressure p The data are measured by changing the average pressure, i.e., the initial pressure of the measurement.
4. Results and discussion 4.1. Relationship of the single-stage Knudsen compressor The performance of the single-stage Knudsen compressor is measured to obtain Eq. (6) for the prediction method in the case = 879, 1240, 1710, of TH = 431 K, TL = 312 K, TR = 328 K, and p chosen in this experiment cor2330, and 3170 Pa. The values of p respond to the saturated vapor pressures of water at 278, 283, 288, 293, and 298 K, respectively. Since this experiment is always conducted at pressures lower than the saturated vapor pressure of TH, TL, and TR, water vapor does not condensate in the apparatus, and the pressure change in the chambers is only due to the Knudsen compressor transport effect. The gas constant of water vapor is 462 J/(kgK). Fig. 5 shows the time history of the pressure difference (red = 879, 1710, and dots) and its fitting curve (black line) for p 3170 Pa. Under each condition, the transient pressure difference is well fitted by the exponential function. The maximum pressure _ MAX appear at difference DpMAX and the maximum mass flow rate m _ MAX , and s _ ¼ 0 and Dp = 0, respectively. The values of DpMAX, m m for each condition are shown in Table 1. Fig. 6 shows the relation and DpMAX and Fig. 7 shows the relationship ship between p and m _ MAX . In this measured range, data points are between p
50
ð18Þ
Thus, by measuring the time variation of the pressure difference _ p ; DpÞ can be obtained. generated by the Knudsen compressor, mð _ MAX for p are derived by meaThe functional forms of DpMAX and m _ MAX for several values of p expersuring the values of DpMAX and m imentally and fitting these values with appropriate functions. 3.4.2. Measuring the performance of the Knudsen compressor The Knudsen compressor and measurement setup used in this study are explained in detail elsewhere [22]. The Knudsen compressor unit is composed of a heater, a cooler, and a porous
= 3170 Pa
40
(Pa)
88
= 1710 Pa
30
= 879 Pa
20 10 0 0
20
40
t (s) Fig. 5. Time history of the pressure difference (red dots) and its fitting curve (black = 879, 1710, and 3170 Pa. (For interpretation of the references to color in line) for p this figure legend, the reader is referred to the web version of this article.)
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K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
4.2. Results of the performance prediction of the heat pump system using the multi-stage Knudsen compressor
Table 1 _ MAX and s for each condition. Measured values of DpMAX, m (Pa) p
DpMAX (Pa)
_ MAX [mg/(scm2)] m
s (s)
879 1240 1710 2333 3170
31.4 36.5 38.3 39.1 40.5
0.0225 0.0260 0.0293 0.0322 0.0339
5.48 5.52 5.13 4.77 4.69
(Pa) MAX
40
20
ðnÞ
ðnÞ
0
ðnÞ
0
2000
ðnÞ
of the temperatures T 0 and T N for each Dt. There is no difference with respect to the transient response and the steady value. From this result, Dt = 5 106 s is used in the following. We derive the areas Ai and the number of stages N necessary to achieve a practical heat pump performance by the prediction method. Fig. 9 shows the predicted heat pump performance in the cases of (Ai, N) = (2.00 m2, 30), (4.00 m2, 15), and (4.00 m2, 30). Fig. 9a and b represents the time histories of the pressures
4000
(Pa) and DpMAX with polynomial curve fitting. Fig. 6. Relationship between p
ðnÞ
ðnÞ
ðnÞ
ðnÞ
p0 and pN and temperatures T 0 and T N in space(0) and space (N), respectively. The time to reach the steady state was shortened by increasing the areas Ai, and the pressure difference ðnÞ ðnÞ ðnÞ ðnÞ DpðnÞ ¼ pðnÞ ¼ TN T0 N p0 and the temperature difference DT in the steady state were increased by increasing the number of
0.04
[mg/(s·cm2)]
ðnÞ
sient temperatures T 0 and T N in the cases of Dt = 2 106 and 5 106 s. Fig. 8 shows the predicted results of the time histories
10
MAX
ð0Þ
pressure in each space pi is uniform with the saturated vapor pressure of that temperature. In this case, any pi remains inside the measured range during the calculations. The other parameters of the 1D model are set as follows: V0 = VN = 100 cm3, Vi = 10 cm3 (i = 1, 2, . . . , N 1), TM = 372 K, and C0 = CN = 100 J/K. Let the latent heat of water L be constant at 2400 kJ/kg regardless of temperature dependence. The dependence of Dt in this calculation is checked by the tran-
50
30
First, the initial conditions are given as follows: the temperatures in space(0) and space(N) are equal at TInit = 285 K, and the
ðnÞ
stages N. In the steady state, p0 was reduced by 455 Pa compared
0.03
to the initial pressure, and
ðnÞ pN
was increased by 638 Pa in the case ðnÞ
2
of (Ai, N) = (4.00 m , 30). On the other hand, T 0 decreased by 5.87
0.02
K with respect to the initial temperature, and T N increased by 5.86 K. Fig. 9c represents the time history of the output power
0.01
Q 0 in space(0), which is derived by Eq. (8) with the value of _ ðnÞ . Q ðnÞ was maximized at t = 0 because there was no pressure M p;0 0 difference, and decreased as pressure difference increased with time. As the number of stages decreases, the decreasing gradient
0
ðnÞ
ðnÞ
0
2000
4000
ðnÞ
(Pa)
of Q 0 becomes larger, and when the area decreases, the initial output power becomes smaller. The initial output power was 2.59 kW
and m _ MAX with polynomial curve fitting. Fig. 7. Relationship between p
ðnÞ
in the case of (Ai, N) = (4.00 m2, 30). The output power Q N in space ðnÞ Q0 .
. Here, DpMAX and m _ MAX are fitsmoothly aligned with respect to p ; quadratic polynomials are ted with appropriate functions of p used in this study, as in Eqs. (19) and (20):
2 p p _ MAX ðp Þ ¼ b0 þ b1 m þ b2 ; Ref Ref p p
295
ð19Þ
ð20Þ
Ref (Pa) are constants. where ai (Pa), bi [mg/(scm2)] (i = 0, 1, 2), and p Here, a0 = 2.28 101 Pa, a1 = 1.29 101 Pa, a2 = 2.35 100 Pa, b0 = 1.24 102 mg/(scm2), b1 = 1.35 102 mg/(scm2), b2 = 3 2 2.12 10 mg/(scm ), and pRef = 1000 Pa. These quadratic functions have small errors for actual data in the measured range of = 879–3170 Pa, with the standard errors being sufficiently small p _ MAX is 1.37 104 mg/(scm2), but that DpMAX is 1.37 Pa and m might contain large errors outside of the measurement range. The mass flow rate can be functionalized with respect to the average pressure and pressure difference by substituting Eqs. (19) and (20) into Eq. (18).
(K)
2 p p þ a2 ; Ref Ref p p
290 285
,
Þ ¼ a0 þ a1 DpMAX ðp
(N) was almost the same as This can be seen from Fig. 9b that the temperatures varied symmetrically. On the other hand, the pressure variation occurred asymmetrically because the saturated
t=2
10-6 s
t=5
10-6 s
280 275 0
10
t(n) ðnÞ
20
(s) ðnÞ
Fig. 8. Time histories of the temperatures T 0 and T N are shown. The blue dashed and red solid lines correspond to the cases of Dt = 2 106 and 5 106 s, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
90
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
(a)
(b) 295
(K)
(Pa)
2500 2000 1500
290
,
,
285
1000
280
500 0
10
t(n)
0
10
(s)
20
t(n) (s) (d)
3 2
3
(kW)
(kW)
(c)
275
20
≥1 kW, ≥6 K
2 1
1
0
0 0
10
t(n)
20
0
5
10
15
(K)
(s)
(Ai, N) = (2.00 m2, 30) (Ai, N) = (4.00 m2, 15) (Ai, N) = (4.00 m2, 30) ðnÞ
ðnÞ
Fig. 9. Predicted heat pump performance in the cases of (Ai, N) = (2.00 m2, 30), (4.00 m2, 15), and (4.00 m2, 15). (a) Time histories of the pressures p0 and pN in space(0) and ðnÞ ðnÞ ðnÞ space(N). (b) Time histories of the temperatures T 0 and T N . in space(0) and space(N). (c) Time history of the output power Q 0 in space(0). (d) Relationship between the temperature difference DT and the output power Q0.
vapor pressure curve was not linear as shown in Fig. 3. Fig. 9d represents the relationship between the temperature difference DT and the output power Q0 derived by eliminating t, i.e., the performance of the heat pump system. Table 2 shows the predicted values of DTMAX and Q0MAX for each condition. From Fig. 9d, the relationship between the output power and the temperature difference was found to be as follows:
DT : Q 0 ¼ Q 0MAX 1 DT MAX
ð21Þ
Let the practical temperature difference be DT = 6.00 K and the practical output power be Q0 = 1.00 kW. As shown in Fig. 9d, it is impossible to simultaneously satisfy DT 6 K and Q0 1 kW in the cases of (Ai, N) = (2.00 m2, 30) and (4.00 m2, 15). In the case of (Ai, N) = (4.00 m2, 30), the output power becomes 1.27 kW at DT = 6.00 K, which is larger than 1.00 kW. It is found that practical heat pump performance (1 kW, 6 K) can be achieved by a 30-stage Knudsen compressor using glass-fiber filters with an area of 4.00 m2. Ai and N will become smaller as the single-stage Knudsen compressor performance improves in the future by optimizing the porous material and heating and cooling methods.
Table 2 Predicted values of DTMAX and Q0MAX for each condition. Ai (m2)
N
DTMAX (K)
Q0MAX (kW)
2.00 4.00 4.00
30 15 30
11.7 5.91 11.7
1.30 2.59 2.59
In this method, the heat pump performance is predicted from transient response simulations. In the transient state, the vapor flow by the Knudsen compressor causes the pressure variation as shown in Fig. 9a along with evaporation and condensation in the evaporator and condenser. Meanwhile, the heat pumps are practically often used with a constant temperature difference maintained between the evaporator and condenser. In such a steady state, the flow contributes only to evaporation and condensation. The contribution to the pressure variation was calculated to be negligibly small in comparison with that to evaporation and condensation in this simulation. Thus, the output power obtained by this method is reasonable to consider as that in a steady state. One foreseen difficulty for fabricating an operational prototype of this heat pump is the design of the evaporator and the condenser. Another is magnification of the size of the Knudsen compressor. To demonstrate the operation of the heat pump, a large-sized Knudsen compressor with an output power that exceeds the amount of heat loss from the measurement device is required. 5. Conclusion A novel heat pump system was proposed using a multi-stage Knudsen compressor with porous materials in the cycle between the evaporator and condenser. A 1D model for the performance prediction of this heat pump was constructed to clarify the specifications for generating a practical output power and temperature difference (1 kW, 6 K). This 1D model is based on the model that accurately predicts the gas transport by the multi-stage
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
Knudsen compressor proposed in a previous paper [22], and evaporation and condensation phenomena and the time variation of the temperatures in the evaporator and condenser are newly added. We derived the configuration of the multi-stage Knudsen compressor necessary to achieve practical heat pump performance. It was shown that a heat pump system with a 30-stage Knudsen compressor using glass-fiber filters with an area of 4.00 m2 could steadily operate at an output power of 1.27 kW and a temperature difference of 6.00 K. Conflicts of interest There are no conflicts of interest to declare. References [1] Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Birkhäuser, Boston, 2007. [2] M. Knudsen, Eine Revision der Gleichgewichtsbedingung der Gase. Thermische Molekularströmung, Ann. Phys. 31 (1909) 205. [3] M. Knudsen, Thermischer Molekulardruck der Gase in Röhren, Ann. Phys. 33 (1910) 1435. [4] C. Cercignani, A. Daneri, Flow of a rarefied gas between two parallel plates, J. Appl. Phys. 34 (1963) 3509. [5] T. Ohwada, Y. Sone, K. Aoki, Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules, Phys. Fluids A 1 (1989) 2042. [6] F. Sharipov, G. Bertoldo, Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number, Phys. Fluids 21 (2009) 067101. [7] C.C. Chen, I.K. Chen, T.P. Liu, Y. Sone, Thermal transpiration for the linearized Boltzmann equation, Commun. Pure Appl. Math. 60 (2007) 147. [8] S.K. Loyalka, Thermal transpiration in a cylindrical tube, Phys. Fluids 12 (1969) 2301–2305. [9] H. Niimi, Thermal creep flow of rarefied gas between two parallel plates, J. Phys. Soc. Jpn. 30 (1971) 572–574. [10] Y. Sone, K. Yamamoto, Flow of rarefied gas through a circular pipe, Phys. Fluids 11 (1968) 1672–1678.
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[11] J. Liu, N.K. Gupta, K.D. Wise, Y.B. Gianchandani, X. Fan, Demonstration of motionless Knudsen pump based micro-gas chromatography featuring microfabricated columns and on-column detectors, Lab Chip 11 (2011) 3487. [12] Y. Qin, Y.B. Gianchandani, Flow of rarefied gas through a circular pipe, Microsyst. Nanoeng. 2 (2016) 15049. [13] N.K. Gupta, Y.B. Gianchandani, Thermal transpiration in mixed cellulose ester membranes: enabling miniature motionless gas pumps, Micropor. Mesopor. Mater. 142 (2011) 535. [14] S. An, N.K. Gupta, Y.B. Gianchandani, A Si-micromachined 162-stage two-part Knudsen pump for on-chip vacuum, J. Microelectromech. Syst. 23 (2014) 406. [15] S. Nakaye, H. Sugimoto, Demonstration of a gas separator composed of Knudsen pumps, Vacuum 125 (2016) 154–164. [16] S.E. Vargo, E.P. Muntz, G.R. Shiflett, Knudsen compressor as a micro- and macroscale vacuum pump without moving parts or fluids, J. Vac. Sci. Technol. A 17 (1999) 2308. [17] M. Young, Y.L. Han, E.P. Muntz, G.R. Shiflett, Characterization and optimization of a radiantly driven multi-stage Knudsen compressor, in: Rarefied Gas Dynamics, AIP, Melville, 2005, p. 174. [18] K. Pharas, S. McNamara, Knudsen pump driven by a thermoelectric material, J. Micromech. Microeng. 20 (2010) 125032. [19] S. McNamara, Y.B. Gianchandani, On-chip vacuum generated by a micromachined Knudsen pump, J. Microelectromech. Syst. 14 (2005) 741. [20] Y. Hirota, R. Iwata, T. Yamauchi, M. Orihashi, M. Morita, Development of a Compact Adsorption Heat Pump System for Automotive Air Conditioning System, SAE Technical Paper 2016-01-0181, 2016. [21] K. Kugimoto, Y. Hirota, Y. Kizaki, Motionless heat pump – a new application of thermal transpiration, in: H. Struchtrup (Ed.), Rarefied Gas Dynamics, AIP, Melville, 2016, p. 080004. [22] K. Kugimoto, Y. Hirota, Y. Kizaki, H. Yamaguchi, T. Niimi, Performance prediction method for a multi-stage Knudsen pump, Phys. Fluids 29 (2017) 122002. [23] R.C. Rodgers, G.E. Hill, Equations for vapour pressure versus temperature: derivation and use of the Antoine equation on a hand-held programmable calculator, Br. J. Anaesth. 50 (1978) 415. [24] M. Rojas-Cárdenas, I. Graur, P. Perrier, J.G. Méolans, Time-dependent experimental analysis of a thermal transpiration rarefied gas flow, Phys. Fluids 25 (2013) 072001. [25] H. Yamaguchi, P. Perrier, M.T. Ho, J.G. Méolans, T. Niimi, I. Graur, Mass flow rate measurement of thermal creep flow from transitional to slip flow regime, J. Fluid Mech. 795 (2016) 690–707. [26] H. Yamaguchi, T. Hanawa, O. Yamamoto, Y. Matsuda, Y. Egami, T. Niimi, Experimental measurement on tangential momentum accommodation coefficient in a single microtube, Microfluid. Nanofluid. 11 (2011) 57–64.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A novel heat pump system using a multi-stage Knudsen compressor K. Kugimoto a,b,⇑, Y. Hirota a, T. Yamauchi a, H. Yamaguchi b, T. Niimi b a b
Toyota Central R&D Labs., Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan Department of Micro-Nano Mechanical Science and Engineering, Nagoya University, Chikusa Furo-cho, Nagoya, Aichi 464-8603, Japan
a r t i c l e
i n f o
Article history: Received 9 March 2018 Received in revised form 14 June 2018 Accepted 14 June 2018
Keywords: Microflow Heat recovery High Knudsen number Thermal molecular pump Thermal transpiration Performance prediction
a b s t r a c t A novel heat pump system is proposed using a multi-stage Knudsen compressor in the cycle between the evaporator and condenser. The proposed Knudsen heat pump is driven by thermal energy, and it is able to utilize waste heat. There are no moving parts in a Knudsen compressor, leading to several advantages, including a lack of vibration and noise, and high durability. In this study, the configuration necessary to achieve a performance such that it can be used as a practical heat pump (output power 1 kW, temperature difference 6 K) is considered. The performance of this heat pump is predicted by a onedimensional analytical model coupled with a simple experimental result. This method is based on our previously constructed method to predict the performance of a multi-stage Knudsen compressor, and the evaporation and condensation and the time variations of temperature in the evaporator and condenser are newly considered. A heat pump with a 30-stage Knudsen compressor using glass-fiber filters with an area of 4.00 m2 is predicted to generate an output power of 1.27 kW and a temperature difference of 6.00 K. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction In a rarefied gas regime, a temperature gradient of a channel wall induces a flow, which is called a thermal transpiration [1]. Knudsen initially observed experimentally the phenomenon at low pressures [2,3], and many studies were performed for academic interest after the first report [4–10]. Recently, the utilization of microporous materials and microfabrication technology has enabled the generation of thermal transpiration at high pressures, such as at atmospheric pressure. This phenomenon is utilized to realize a pump/compressor, which is called a Knudsen pump/compressor. A Knudsen compressor is driven by thermal energy, and it is able to utilize waste heat or environmental heat energy as a driving source, so it is expected to be a new class of heat recovery device. In addition, it has various advantages, such as a lack of vibration and noise, high durability, and easy miniaturization compared to other existing compressors because it has no mechanically moving parts. Several practical applications of a Knudsen pump/compressor have been proposed [11–19]. For example, Liu et al. and Qin and Gianchandani investigated a micro-gas chromatography system ⇑ Corresponding author at: Toyota Central R&D Labs., Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan. E-mail addresses: [email protected] (K. Kugimoto), e1476@mosk. tytlabs.co.jp (Y. Hirota), [email protected] (T. Yamauchi), hiroki@ nagoya-u.jp (H. Yamaguchi), [email protected] (T. Niimi). https://doi.org/10.1016/j.ijheatmasstransfer.2018.06.072 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
with a miniature Knudsen pump [11,12]. Nakaye and Sugimoto demonstrated a gas separator composed of Knudsen pumps [15]. However, applications of Knudsen pumps/compressors are still limited. One engineering product that uses a gas compressor is a heat pump. A heat pump system performs cooling and heating using the latent heat of a refrigerant, and it is used for air-conditioners, refrigerators, and the like. In general, an electrical compressor is used to generate a pressure difference between the evaporator and condenser so as to circulate the refrigerant vapor and to promote evaporation and condensation of the refrigerant. For heat recovery purposes, a heat-driven gas compressor may be used. An adsorption heat pump (AHP) is one such system [20]. In this system, the refrigerant vapor is sucked from the evaporator by adsorption of the adsorbent, and it is discharged to the condenser by desorption. The adsorbent is cooled with cooling water during adsorption, and it is heated with a waste heat medium during desorption. Therefore, it is necessary for AHP to repeatedly switch the gas flow path and the temperature control liquid flow path of the adsorbent, so many switching mechanisms are required. It may be possible to create a heat pump system with a simpler mechanism for heat recovery purposes by using a Knudsen compressor which can generate a practical pressure difference and mass flow. This study proposes a novel heat pump system using a multistage Knudsen compressor with porous materials as a compressor in the heat pump cycle, with water vapor as a refrigerant. The
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
transport of water vapor by a compressor was verified by our previous measurements [21]. However, the throughput of the Knudsen compressor is known to be low compared with a conventional mechanical compressor in a meter-scale system. Therefore, we predict the performance of the proposed heat pump system and clarify the configuration necessary to achieve the practical output power and temperature difference of conventional systems (e.g., 1 kW, 6 K). This prediction is made using the method for the multi-stage Knudsen compressor proposed in our previous study [22], with some refinements. This method was based on the one-dimensional analytical model from the mass conservation law for a multi-stage Knudsen compressor combined with a simple experimental result for a single stage, aiming to simulate a realistic performance of a high flow rate Knudsen compressor using a porous material with complicated aggregate channels. In this study, the prediction method is modified by adding the evaporation/condensation and the temperature time variations of the evaporator/ condenser. The output power of the proposed heat pump depends on the mass flow rate generated by the Knudsen compressor, while the temperature difference depends on the pressure difference. Therefore, the performance is evaluated to clarify the area of the porous material and the number of stages necessary to achieve a practical performance of an output power 1 kW and a temperature difference 6 K. 2. Heat pump system using a multi-stage Knudsen compressor 2.1. Knudsen compressor The degree of rarefaction of gas is expressed by the Knudsen number Kn. The Knudsen number of a gas in a channel is defined by the ratio of the mean free path of gas l to the characteristic length of the channel D.
Kn ¼
l : D
ð1Þ
The gas whose Kn is larger than 0.1 is considered to be a rarefied gas. Under this condition, a one-way gas flow is induced by the temperature difference between the two ends of the channel from the cold to hot side, which is called the thermal transpiration. (a) Microchannel
Kn>0.1
(b)
Thick channel
Kn<<0.1
Microchannel Connection part
85
An illustrative single-stage Knudsen compressor is shown in Fig. 1a. A single-stage Knudsen compressor is composed of a microchannel having a diameter DP of the same order of magnitude or less compared with the mean free path of gas and a channel having a diameter DC sufficiently larger than the mean free path. When given a mountain-shaped temperature gradient with a high temperature TH and a low temperature TC as in the lower figure of Fig. 1a, thermal transpirations are induced in both of the microchannel and thick channel, but a pressure-gradient driven counterflow results in no net flow only in the thick channel. Thereby, the Knudsen compressor can generate a one-way flow and a pressure difference between the inlet and outlet of the compressor, although there is no temperature difference between the inlet and outlet. Then, a multi-stage Knudsen compressor can be built by connecting multiple single-stage Knudsen compressors in series as in Fig. 1b. The possible pressure difference increases as the number of stages increases, while the mass flow rate is determined by the performance of the single stage. 2.2. Novel heat pump system The configuration of our novel heat pump system is shown in Fig. 2. Two chambers, referred to hereinafter as the evaporator and condenser, are filled with liquid water and water vapor as a refrigerant. A heat exchanger is installed within each chamber in order to take the heat inside the system to the outside. The evaporator and condenser are connected to each other by two channels to form a cycle: one channel in a vapor phase consists of the multistage Knudsen compressor, and the other in a liquid phase has a valve. When the Knudsen compressor is operated, vapor is made to flow by means of the compressor from the evaporator to the condenser. In the evaporator, the pressure drop due to the flow induced by the compressor promotes water evaporation and heat absorption due to the latent heat of water. Then, the temperature in the evaporator TE decreases corresponding to the decrease in the pressure in the evaporator PE from the saturated vapor pressure curve of water in Fig. 3, which is a monotonically increasing function. In contrast, in the condenser, the pressure increase promotes vapor condensation and heat generation. Then, the increase in the pressure in the condenser PC induces the increase in the temperature in the condenser TC. When the evaporator water decreases, the valve at the channel in a liquid phase is opened and liquid water is supplied from the condenser. As a result, it is possible to operate continuously in a completely isolated state from the outside, and there is no concern that dust will clog microchannels inside the Knudsen compressor. From the operating mechanism, the performance of the heat pump system is determined by the performance of the Knudsen compressor being used. The temperature difference obtained in this heat pump system depends on the pressure difference generated by the Knudsen compressors. The pressure difference can be amplified by cascading compressor units [14]. On the other hand, _ the power output Q depends on the mass flow rate of vapor M. _ In terms of the latent heat of water, L, Q is calculated as Q = LM. 3. Performance prediction method 3.1. Outline
Fig. 1. (a) Illustrative single-stage Knudsen compressor. (b) Illustrative multi-stage Knudsen compressor. The pressure difference is amplified by combining multiple single-stage Knudsen compressors in series.
The performance is predicted by modifying the prediction method for the multi-stage Knudsen pump/compressor in our previous study [22], where the practical performance was well predicted by the 1D analytical model based on the mass conservation law using the single-stage experimental results. The
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K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
Water vapor
Evaporator
Condenser Multi-stage Knudsen compressor
Water vapor
Heat exchanger
Heat exchanger
Liquid water
Valve
Saturated vapor pressure (Pa)
Fig. 2. Device configuration of the heat pump system using a multi-stage Knudsen compressor.
8000
3.2. 1D model
6000
3.2.1. Derivation of time evolution recurrence formula of pressure distribution A schematic for the performance prediction model of the heat pump system using an N-stage Knudsen compressor is shown in Fig. 4. The mass conservation law applies to N + 1 spaces, which consist of two chambers and N 1 connection parts of the N-stage Knudsen compressor. A temperature difference is imposed on each Knudsen compressor unit so that the left side of the unit is at low temperature TL and the right side is at high temperature TH. The ith compressor unit from the cold side (left) is called ‘‘compressor(i)”. The space on the left of compressor(i) is called ‘‘space(i 1)”, and the space on the right is called ‘‘space(i)”; namely, the evaporator and condenser correspond to space(0) and space(N), respectively. Let Ai and Vi be the cross-sectional area of the porous material in compressor(i) and the volume of space(i), respectively. The time t(n) at the nth step is expressed using the time step Dt as ðnÞ ðnÞ _ ðnÞ are the pressure and average temt(n) = nDt. Let p , T , and M
4000
PC
2000
PE
0 273
TC
TE
293
313
Temperature (K) Fig. 3. Saturated vapor pressure curve of water [23]. Examples of PE, PC, TE, and TC are shown (steady-state values in calculations performed in Section 4.2).
water vapor is transported by the multi-stage Knudsen compressor, and evaporation in the evaporator and condensation in the condenser are newly considered in this study. The time histories of the pressure distribution inside the heat pump and temperatures in the evaporator and condenser are simulated.
Space(i-1) ( pi(−n1) , Vi −1, TM )
i
(n) 0
p;i
Space(i) ( pi( n ), Vi , TM )
Compressor(1) (Porous area: A1 , Compressor(i) Compressor(N) ( n) (AN , M p( n, N) ) Mass flow rate: M p( n,1) ) (Ai , M p ,i )
Space(0) Pressure: p
i
perature in space(i) at time t(n) and the mass flow rate generated by
Space(N)
p N(n ) VN
Volume: V0
TN(n )
(n)
Temperature: T0
(n)
Evaporation rate: M p , 0 Heat capacity: C0
Condensation rate:M p( n, N) +1
TL TH
CN
Fig. 4. Schematic for the performance prediction model of the heat pump system using an N-stage Knudsen compressor.
87
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91 ðnÞ
ðnÞ
compressor(i), respectively. T 0 and T N correspond to TE and TC, respectively. As for average temperature, it is simply referred to as ‘‘temperature” below. Different from the mass flow rate in other _ ðnÞ are mass flows generated by evaporation of _ ðnÞ and M spaces, M
3.2.2. Relational expression of pressure, temperature, evaporation and condensation rate in space(0) and space(N) There are three unknown values in space(0) and space(N), the pressure, temperature, and evaporation (or condensation) amount.
water in the evaporator and condensation of water vapor in the condenser, respectively. The temperature in the space(1), . . . ,
Equation (7) shows the relationship between the pressure p0
p;0
p;Nþ1
space(N 1) is assumed to be constant and ðnÞ
ðnÞ Ti
= TM = (TL + TH)/2
ðnÞ
(i = 1, 2, . . . , N 1); however, T 0 and T N are variables in this study. Heat capacities in space(0) and space(N) are denoted by C0 and CN. It is assumed that the heat capacity of the container is sufficiently larger than the heat capacity of water, and C0 and CN are fixed with respect to time. Using the mass of the gas in space(i) at the nth step and its timeðnÞ
ðnÞ
ðnþ1Þ
varying rate be Mc;i and dMc;i =dt, respectively, the mass M c;i th
ðnþ1Þ
M c;i
ðnÞ
ðnÞ
¼ M c;i þ Dt
dM c;i dt
ði ¼ 0; 1; . . . ; NÞ:
ð2Þ
The mass M c;i of the gas is obtained from the density qi ðnÞ
ðnÞ
the volume Vi, and q
ðnÞ i
ðnÞ
M c;i ¼ qi V i ¼ ðnÞ
ðnÞ pi ðnÞ RT i
Vi
and
is expressed as follows using the gas conðnÞ
and pressure pi :
stant of water vapor R, temperature T i ðnÞ
at
step can be expressed by the following equation:
the n + 1
ð3Þ
_ ðnÞ from space(i): space(i) and the outflow rate M p;iþ1
ði ¼ 0; 1; . . . ; NÞ:
ð4Þ
_ ðnÞ equation using mass flow rate per unit area m p;i and Ai:
Here,
ðnÞ _ p;i m
ði ¼ 1; 2; . . . ; NÞ:
ð5Þ
ðnÞ
ðnÞ
ðnÞ
difference across compressor(i) Dpp;i ¼ pi1 þ pi , determined from the experimental results of a single-stage Knudsen compressor, which is described later, in Section 3.4.1. ðnÞ _ ðnÞ ðnÞ _ ðnÞ ði ¼ 1; 2; . . . ; NÞ: m p;i ¼ mp;i ðpp;i ; Dpp;i Þ
ð6Þ
By substituting Eqs. (3) and (4) into Eq. (2), the pressure at the n + 1th step can be calculated as follows: ðnþ1Þ
ðnþ1Þ
¼
¼
_ ðnÞ LM p;i
Ti
ðnÞ
Ti
ðnÞ
pi þ Dt
ðnþ1Þ
RT i Vi
ð7Þ However, Eq. (7) is not closed, because the evaporation and conden_ ðnÞ , respectively, and the _ ðnÞ and M sation amount per unit time, M p;0
ðnþ1Þ
ðnþ1Þ
p;Nþ1
and T N are unknown. In order to close this temperatures T 0 equation, these relational expressions are further needed. The method of determining these values is described in the next section, Section 3.2.2.
ð8Þ
ðnÞ
= Q N in the steady state, but these values may differ tranðnþ1Þ
siently. The temperatures T 0
ðnþ1Þ
and T N
at the n + 1th step are ðnÞ
ðnÞ
expressed by the following equation using Q 0 and Q N . ðnþ1Þ
¼ T 0 Dt
ðnþ1Þ
¼ T N þ Dt
TN
ðnÞ
ðnÞ
ðnÞ
Q0 ; C0
ð9Þ
ðnÞ
QN : CN
ð10Þ
Substituting Eq. (8) into Eqs. (9) and (10), the relational expressions of the temperature, evaporation, and condensation amount are obtained by the following equations. ðnþ1Þ
T0
ðnþ1Þ
TN
_ LM p;0 ðnÞ
ðnÞ
¼ T 0 Dt
ðnÞ
¼ T N þ Dt
C0
;
ð11Þ
_ ðnÞ LM p;Nþ1 CN
:
ð12Þ
Also, the pressure and temperature in space(0) and space(N) are related by the saturated vapor pressure curve of water. According to Antoine’s equation [23], these relationships are as follows:
log10 ðjpi
ðnþ1Þ
Þ¼ a
b ðnþ1Þ
Ti
c
ði ¼ 0; NÞ:
ð13Þ
Here, j, a, b, and c are constants which depend on substances and units. In the case of water vapor, when using Pa as the unit of and K as the unit of temperature T i , j = 103 pressure pi 1 0 3 1 Pa , a = 7.17 10 , b = 1.72 10 K, and c = 4.06 10 K. By solving these simultaneous equations, the evaporation and condensation amounts per unit time at the nth step and the pressure and temperature at the n + 1th step can be derived in space(0) and space(N). ðnþ1Þ
_ ðnÞ M _ ðnÞ Þ ði ¼ 0; 1; . . . ; NÞ: ðM p;i p;iþ1
ðnÞ
p;Nþ1
ði ¼ 0; NÞ;
is a function of the average pressure between the inlet
ðnÞ ðnÞ ðnÞ p;i and outlet of compressor(i) p ¼ ðpi1 þ pi Þ=2 and the pressure
pi
ðnÞ Qi
T0
Assuming that the surface temperature of each porous material _ ðnÞ (i = 1, 2, . . . , N) can be obtained by the following is uniform, M p;i
ðnÞ _ ðnÞ ¼ m _ p;i M Ai p;i
ðnÞ
The heat absorption and generation per unit time, Q 0 and Q N , are expressed by the following equation with the latent heat of _ ðnÞ . _ ðnÞ and M water L using M
ðnÞ
th
p;i
_ ðnÞ _ ðnÞ M ¼M p;i p;iþ1
and temperature at the n + 1 step and the evaporation _ ðnÞ at the n step (or pðnþ1Þ , T ðnþ1Þ and M _ ðnÞ ). If there amount M N N p;0 p;Nþ1 are four additional independent equations that describe these relationships (two for each of i = 0, N), the numbers of unknowns and equations become equal, resulting in a solvable problem. Here, the relationship between the temperature and the evaporation or the condensation amount is derived by assuming that the evaporator and condenser are completely insulated, and the changes in temperature there are only due to the latent heat of the evaporation and condensation. In addition, the relationship between the pressure and temperature is derived by assuming that the evaporator and condenser are designed to be sufficiently large. Under this assumption, the pressures in space(0) and space(N) are always equal to the saturated vapor pressure of the temperatures there.
Q0
ðnÞ dM c;i =dt
at space(i) in the n _ ðnÞ to step is expressed by the difference between the inflow rate M
dt
th
p;0
ði ¼ 0; 1; . . . ; NÞ:
The time-varying rate of the mass
ðnÞ dM c;i
ðnþ1Þ
ðnþ1Þ T0 th
ðnþ1Þ
3.3. Numerical solution of simultaneous equations by Newton’s method ðnþ1Þ ðnþ1Þ _ ðnÞ (or pðnþ1Þ , T ðnþ1Þ , and M _ ðnÞ ) can be p0 , T 0 , and M N N p;0 p;Nþ1 obtained by simultaneously solving Eqs. (7) for i = 0, (11) and
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
(13) for i = 0 (or Eqs. (7) for i = N, (12) and (13) for i = N). However, algebraic analysis is difficult because Eqs. (7) and (13) are nonlin_ ðnÞ ) is numerically computed by Newton’s _ ðnÞ (or M ear. Thus, M p;0
p;Nþ1
method. _ ðnÞ is determined by the The analytic function f for solving M p;0 following equation using Eq. (7). ðnþ1Þ
ðnþ1Þ
_ ðnÞ ;pðnþ1Þ ;T ðnþ1Þ Þ ¼ pðnþ1Þ þ T 0 pðnÞ þ Dt RT 0 f ðM p;0 0 0 0 0 ðnÞ V0 T0
_ ðnÞ M _ ðnÞ Þ ¼ 0: ðM p;0 p;1 ð14Þ
ðnþ1Þ p0
_ ðnÞ M p;0
ðnþ1Þ T0
Here, and can be expressed as a function of from Eqs. (11) and (13). Thus, f can also be expressed as a function of _ ðnÞ . After that, the value of M _ ðnÞ is obtained by Newton’s method. M p;0
p;0
_ ðnÞ is determined, pðnþ1Þ and T ðnþ1Þ can be easily obtained Once M p;0 0 0 ðnþ1Þ ðnþ1Þ _ ðnÞ can be calculated from Eqs. (11) and (13). pN , T N , and M p;Nþ1 by the same process. 3.4. Relationship between mass flow rate and pressure difference of the single-stage Knudsen compressor 3.4.1. Performance of the Knudsen compressor The mass flow rate of the single-stage Knudsen compressor as a function of the average pressure between the inlet and outlet of the compressor and the pressure difference between them, as in Eq. (6), is derived as follows. Consider a system with a single-stage Knudsen compressor between two chambers with volume V. The time-dependent pressure difference between the chambers generated by the compressor is Dp, which is approximated by the following exponential function of t [24,25]:
t DpðtÞ ¼ DpMAX 1 es ;
ð15Þ
where DpMAX is the pressure difference at steady state, and s is a constant determined by comparing the experimental data and the _ can be derived fitting curve. The mass flow rate per unit area, m, in terms of the pressure difference between the two chambers by the constant volume method as follows [26]:
_ ¼ m
V d Dp ; RT R A dt 2
ð16Þ
where R is the gas constant of the gas in the compressor, TR is the room temperature, and A is the area of the porous material. The _ MAX , appears at t = 0 and Dp = 0 as maximum mass flow rate, m follows:
_ MAX ¼ m
V 1 Dp : 2RT R A MAX s
ð17Þ
_ be a By eliminating V, R, TR, and A from Eqs. (15)–(17), we let m _ MAX as follows: function of Dp with DpMAX and m
_ p ; DpÞ mð Dp ¼1 : Þ Þ _ MAX ðp DpMAX ðp m
material with micropores. The compressor unit is clamped by vacuum flanges with an insertion of square rings. Chilled water at TL or heated oil at TH circulates through the cooler or heater to keep the temperature constant. There are slits, which are the channels for gas, across the cooler and heater, so that the temperature of gas equilibrates to that of the cooler or heater. A glass-fiber filter used as the porous material is glued onto the cooler. The average pore size, thickness, and area of the glass fiber filter are 0.7 lm, 380 lm, and 4.90 cm2, respectively. To measure the performance of the single-stage Knudsen compressor, the single-stage Knudsen compressor is connected between two chambers with volumes of 5400 cm3. A closed loop is formed by connecting the two chambers with the Knudsen compressor and another pipe with a valve. The pressure of each chamber is measured by capacitance manometers (Setra 730G100T (F.S.: 13.3 kPa), 730G-010T (F.S.: 1.33 kPa), accuracy: ±0.150% of F.S.), which remain always connected to the chambers. The performance of the single-stage Knudsen compressor is evaluated by the constant volume method, which measures the time history of the pressure difference between these chambers as generated by the compressor. By closing a valve inserted into a pipe, the time-dependent pressure difference between the chambers, . Eq. (15), is recorded for a certain value of the average pressure p The data are measured by changing the average pressure, i.e., the initial pressure of the measurement.
4. Results and discussion 4.1. Relationship of the single-stage Knudsen compressor The performance of the single-stage Knudsen compressor is measured to obtain Eq. (6) for the prediction method in the case = 879, 1240, 1710, of TH = 431 K, TL = 312 K, TR = 328 K, and p chosen in this experiment cor2330, and 3170 Pa. The values of p respond to the saturated vapor pressures of water at 278, 283, 288, 293, and 298 K, respectively. Since this experiment is always conducted at pressures lower than the saturated vapor pressure of TH, TL, and TR, water vapor does not condensate in the apparatus, and the pressure change in the chambers is only due to the Knudsen compressor transport effect. The gas constant of water vapor is 462 J/(kgK). Fig. 5 shows the time history of the pressure difference (red = 879, 1710, and dots) and its fitting curve (black line) for p 3170 Pa. Under each condition, the transient pressure difference is well fitted by the exponential function. The maximum pressure _ MAX appear at difference DpMAX and the maximum mass flow rate m _ MAX , and s _ ¼ 0 and Dp = 0, respectively. The values of DpMAX, m m for each condition are shown in Table 1. Fig. 6 shows the relation and DpMAX and Fig. 7 shows the relationship ship between p and m _ MAX . In this measured range, data points are between p
50
ð18Þ
Thus, by measuring the time variation of the pressure difference _ p ; DpÞ can be obtained. generated by the Knudsen compressor, mð _ MAX for p are derived by meaThe functional forms of DpMAX and m _ MAX for several values of p expersuring the values of DpMAX and m imentally and fitting these values with appropriate functions. 3.4.2. Measuring the performance of the Knudsen compressor The Knudsen compressor and measurement setup used in this study are explained in detail elsewhere [22]. The Knudsen compressor unit is composed of a heater, a cooler, and a porous
= 3170 Pa
40
(Pa)
88
= 1710 Pa
30
= 879 Pa
20 10 0 0
20
40
t (s) Fig. 5. Time history of the pressure difference (red dots) and its fitting curve (black = 879, 1710, and 3170 Pa. (For interpretation of the references to color in line) for p this figure legend, the reader is referred to the web version of this article.)
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K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
4.2. Results of the performance prediction of the heat pump system using the multi-stage Knudsen compressor
Table 1 _ MAX and s for each condition. Measured values of DpMAX, m (Pa) p
DpMAX (Pa)
_ MAX [mg/(scm2)] m
s (s)
879 1240 1710 2333 3170
31.4 36.5 38.3 39.1 40.5
0.0225 0.0260 0.0293 0.0322 0.0339
5.48 5.52 5.13 4.77 4.69
(Pa) MAX
40
20
ðnÞ
ðnÞ
0
ðnÞ
0
2000
ðnÞ
of the temperatures T 0 and T N for each Dt. There is no difference with respect to the transient response and the steady value. From this result, Dt = 5 106 s is used in the following. We derive the areas Ai and the number of stages N necessary to achieve a practical heat pump performance by the prediction method. Fig. 9 shows the predicted heat pump performance in the cases of (Ai, N) = (2.00 m2, 30), (4.00 m2, 15), and (4.00 m2, 30). Fig. 9a and b represents the time histories of the pressures
4000
(Pa) and DpMAX with polynomial curve fitting. Fig. 6. Relationship between p
ðnÞ
ðnÞ
ðnÞ
ðnÞ
p0 and pN and temperatures T 0 and T N in space(0) and space (N), respectively. The time to reach the steady state was shortened by increasing the areas Ai, and the pressure difference ðnÞ ðnÞ ðnÞ ðnÞ DpðnÞ ¼ pðnÞ ¼ TN T0 N p0 and the temperature difference DT in the steady state were increased by increasing the number of
0.04
[mg/(s·cm2)]
ðnÞ
sient temperatures T 0 and T N in the cases of Dt = 2 106 and 5 106 s. Fig. 8 shows the predicted results of the time histories
10
MAX
ð0Þ
pressure in each space pi is uniform with the saturated vapor pressure of that temperature. In this case, any pi remains inside the measured range during the calculations. The other parameters of the 1D model are set as follows: V0 = VN = 100 cm3, Vi = 10 cm3 (i = 1, 2, . . . , N 1), TM = 372 K, and C0 = CN = 100 J/K. Let the latent heat of water L be constant at 2400 kJ/kg regardless of temperature dependence. The dependence of Dt in this calculation is checked by the tran-
50
30
First, the initial conditions are given as follows: the temperatures in space(0) and space(N) are equal at TInit = 285 K, and the
ðnÞ
stages N. In the steady state, p0 was reduced by 455 Pa compared
0.03
to the initial pressure, and
ðnÞ pN
was increased by 638 Pa in the case ðnÞ
2
of (Ai, N) = (4.00 m , 30). On the other hand, T 0 decreased by 5.87
0.02
K with respect to the initial temperature, and T N increased by 5.86 K. Fig. 9c represents the time history of the output power
0.01
Q 0 in space(0), which is derived by Eq. (8) with the value of _ ðnÞ . Q ðnÞ was maximized at t = 0 because there was no pressure M p;0 0 difference, and decreased as pressure difference increased with time. As the number of stages decreases, the decreasing gradient
0
ðnÞ
ðnÞ
0
2000
4000
ðnÞ
(Pa)
of Q 0 becomes larger, and when the area decreases, the initial output power becomes smaller. The initial output power was 2.59 kW
and m _ MAX with polynomial curve fitting. Fig. 7. Relationship between p
ðnÞ
in the case of (Ai, N) = (4.00 m2, 30). The output power Q N in space ðnÞ Q0 .
. Here, DpMAX and m _ MAX are fitsmoothly aligned with respect to p ; quadratic polynomials are ted with appropriate functions of p used in this study, as in Eqs. (19) and (20):
2 p p _ MAX ðp Þ ¼ b0 þ b1 m þ b2 ; Ref Ref p p
295
ð19Þ
ð20Þ
Ref (Pa) are constants. where ai (Pa), bi [mg/(scm2)] (i = 0, 1, 2), and p Here, a0 = 2.28 101 Pa, a1 = 1.29 101 Pa, a2 = 2.35 100 Pa, b0 = 1.24 102 mg/(scm2), b1 = 1.35 102 mg/(scm2), b2 = 3 2 2.12 10 mg/(scm ), and pRef = 1000 Pa. These quadratic functions have small errors for actual data in the measured range of = 879–3170 Pa, with the standard errors being sufficiently small p _ MAX is 1.37 104 mg/(scm2), but that DpMAX is 1.37 Pa and m might contain large errors outside of the measurement range. The mass flow rate can be functionalized with respect to the average pressure and pressure difference by substituting Eqs. (19) and (20) into Eq. (18).
(K)
2 p p þ a2 ; Ref Ref p p
290 285
,
Þ ¼ a0 þ a1 DpMAX ðp
(N) was almost the same as This can be seen from Fig. 9b that the temperatures varied symmetrically. On the other hand, the pressure variation occurred asymmetrically because the saturated
t=2
10-6 s
t=5
10-6 s
280 275 0
10
t(n) ðnÞ
20
(s) ðnÞ
Fig. 8. Time histories of the temperatures T 0 and T N are shown. The blue dashed and red solid lines correspond to the cases of Dt = 2 106 and 5 106 s, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
(a)
(b) 295
(K)
(Pa)
2500 2000 1500
290
,
,
285
1000
280
500 0
10
t(n)
0
10
(s)
20
t(n) (s) (d)
3 2
3
(kW)
(kW)
(c)
275
20
≥1 kW, ≥6 K
2 1
1
0
0 0
10
t(n)
20
0
5
10
15
(K)
(s)
(Ai, N) = (2.00 m2, 30) (Ai, N) = (4.00 m2, 15) (Ai, N) = (4.00 m2, 30) ðnÞ
ðnÞ
Fig. 9. Predicted heat pump performance in the cases of (Ai, N) = (2.00 m2, 30), (4.00 m2, 15), and (4.00 m2, 15). (a) Time histories of the pressures p0 and pN in space(0) and ðnÞ ðnÞ ðnÞ space(N). (b) Time histories of the temperatures T 0 and T N . in space(0) and space(N). (c) Time history of the output power Q 0 in space(0). (d) Relationship between the temperature difference DT and the output power Q0.
vapor pressure curve was not linear as shown in Fig. 3. Fig. 9d represents the relationship between the temperature difference DT and the output power Q0 derived by eliminating t, i.e., the performance of the heat pump system. Table 2 shows the predicted values of DTMAX and Q0MAX for each condition. From Fig. 9d, the relationship between the output power and the temperature difference was found to be as follows:
DT : Q 0 ¼ Q 0MAX 1 DT MAX
ð21Þ
Let the practical temperature difference be DT = 6.00 K and the practical output power be Q0 = 1.00 kW. As shown in Fig. 9d, it is impossible to simultaneously satisfy DT 6 K and Q0 1 kW in the cases of (Ai, N) = (2.00 m2, 30) and (4.00 m2, 15). In the case of (Ai, N) = (4.00 m2, 30), the output power becomes 1.27 kW at DT = 6.00 K, which is larger than 1.00 kW. It is found that practical heat pump performance (1 kW, 6 K) can be achieved by a 30-stage Knudsen compressor using glass-fiber filters with an area of 4.00 m2. Ai and N will become smaller as the single-stage Knudsen compressor performance improves in the future by optimizing the porous material and heating and cooling methods.
Table 2 Predicted values of DTMAX and Q0MAX for each condition. Ai (m2)
N
DTMAX (K)
Q0MAX (kW)
2.00 4.00 4.00
30 15 30
11.7 5.91 11.7
1.30 2.59 2.59
In this method, the heat pump performance is predicted from transient response simulations. In the transient state, the vapor flow by the Knudsen compressor causes the pressure variation as shown in Fig. 9a along with evaporation and condensation in the evaporator and condenser. Meanwhile, the heat pumps are practically often used with a constant temperature difference maintained between the evaporator and condenser. In such a steady state, the flow contributes only to evaporation and condensation. The contribution to the pressure variation was calculated to be negligibly small in comparison with that to evaporation and condensation in this simulation. Thus, the output power obtained by this method is reasonable to consider as that in a steady state. One foreseen difficulty for fabricating an operational prototype of this heat pump is the design of the evaporator and the condenser. Another is magnification of the size of the Knudsen compressor. To demonstrate the operation of the heat pump, a large-sized Knudsen compressor with an output power that exceeds the amount of heat loss from the measurement device is required. 5. Conclusion A novel heat pump system was proposed using a multi-stage Knudsen compressor with porous materials in the cycle between the evaporator and condenser. A 1D model for the performance prediction of this heat pump was constructed to clarify the specifications for generating a practical output power and temperature difference (1 kW, 6 K). This 1D model is based on the model that accurately predicts the gas transport by the multi-stage
K. Kugimoto et al. / International Journal of Heat and Mass Transfer 127 (2018) 84–91
Knudsen compressor proposed in a previous paper [22], and evaporation and condensation phenomena and the time variation of the temperatures in the evaporator and condenser are newly added. We derived the configuration of the multi-stage Knudsen compressor necessary to achieve practical heat pump performance. It was shown that a heat pump system with a 30-stage Knudsen compressor using glass-fiber filters with an area of 4.00 m2 could steadily operate at an output power of 1.27 kW and a temperature difference of 6.00 K. Conflicts of interest There are no conflicts of interest to declare. References [1] Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Birkhäuser, Boston, 2007. [2] M. Knudsen, Eine Revision der Gleichgewichtsbedingung der Gase. Thermische Molekularströmung, Ann. Phys. 31 (1909) 205. [3] M. Knudsen, Thermischer Molekulardruck der Gase in Röhren, Ann. Phys. 33 (1910) 1435. [4] C. Cercignani, A. Daneri, Flow of a rarefied gas between two parallel plates, J. Appl. Phys. 34 (1963) 3509. [5] T. Ohwada, Y. Sone, K. Aoki, Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules, Phys. Fluids A 1 (1989) 2042. [6] F. Sharipov, G. Bertoldo, Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number, Phys. Fluids 21 (2009) 067101. [7] C.C. Chen, I.K. Chen, T.P. Liu, Y. Sone, Thermal transpiration for the linearized Boltzmann equation, Commun. Pure Appl. Math. 60 (2007) 147. [8] S.K. Loyalka, Thermal transpiration in a cylindrical tube, Phys. Fluids 12 (1969) 2301–2305. [9] H. Niimi, Thermal creep flow of rarefied gas between two parallel plates, J. Phys. Soc. Jpn. 30 (1971) 572–574. [10] Y. Sone, K. Yamamoto, Flow of rarefied gas through a circular pipe, Phys. Fluids 11 (1968) 1672–1678.
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