Theoretical Models for the Cooling Power and Temperature of Dilution Refrigerators
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Transcript of Theoretical Models for the Cooling Power and Temperature of Dilution Refrigerators
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J Low Temp Phys (2010) 158: 901–921
DOI 10.1007/s10909-009-0060-3
Theoretical Models for the Cooling Power and Base
Temperature of Dilution Refrigerators
Patrick Wikus · Tapio O. Niinikoski
Received: 8 July 2009 / Accepted: 23 November 2009 / Published online: 9 December 2009
© Springer Science+Business Media, LLC 2009
Abstract 3He/ 4He dilution refrigerators are widely used for applications requiring
continuous cooling at temperatures below approximately 300 mK. Despite of the
popularity of these devices in low temperature physics, the thermodynamic relations
underlying the cooling mechanism of 3He/ 4He refrigerators are very often incorrectly
used. Several thermodynamic models of dilution refrigeration have been published in
the past, sometimes contradicting each other. These models are reviewed and com-
pared with each other over a range of different 3He flow rates.
In addition, a new numerical method for the calculation of a dilution refrigerator’s
cooling power at arbitrary flow rates is presented. This method has been developed
at CERN’s Central Cryogenic Laboratory. It can be extended to include many effects
that cannot easily be accounted for by any of the other models, including the degrada-
tion of heat exchanger performance due to the limited number of step heat exchanger
elements, which can be considerable for some designs.
Finally, the limitations of applying the results obtained with idealized thermody-
namic models to actual dilution refrigeration systems are discussed.
Keywords Dilution refrigeration · Thermal modeling · Countercurrent heat
exchanger · Cooling power · Base temperature
P. Wikus ()
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, USA
e-mail: [email protected]
T.O. Niinikoski
CERN, 1211 Geneva 23, Switzerland
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1 Introduction
Using the enthalpy of mixing of 3He and 4He for cooling was first suggested by
London, Clarke and Mendoza in 1962 [1]. The first actual dilution refrigerator was
built shortly after in 1965 by Das, de Bruyn Ouboter and Taconis at Leiden University[2].
Radebaugh published a comprehensive text on the thermodynamics underlying3He/ 4He dilution refrigeration, with a compilation of the properties of the working
fluids, in 1967 [3]. In 1976, Niinikoski presented the first thermodynamic model of
dilution refrigerators which permitted to calculate the mixing chamber temperature
as a function of the heat load and heat exchanger surface area at the optimum flow
rate [4]. Frossati and co-workers subsequently developed a different model for ar-
bitrary 3He flow rates [5 and 6], and subsequently modified that model [7]. In 1994,
Takano published an improvement of Frossati’s model [8]. In the following, the terms“Niinikoski’s model”, “Frossati’s model” and “Takano’s model” will be used to refer
to their respective work.
The cooling cycle of a continuously operating dilution refrigerator is discussed in
several contemporary textbooks such as [9] or [10]. Detailed theoretical treatments of
the cooling power and base temperature, which are the subject of this work, are usu-
ally omitted, however. Figure 1 shows a dilution refrigerator which has been designed
by the authors and is currently in operation at CERN’s Central Cryogenic Laboratory.
The cooling power Q̇m available in the mixing chamber, at a 3He flow rate ṅ3, is
given by the energy balance of the mixing chamber
Q̇m = ṅ3 · (H l (T m) − H c(T o)), (1)
where H l (T m) is the enthalpy of 3He in saturated dilute solution at the fluid temper-
ature T m in the mixing chamber, and H c(T o) is the enthalpy of the concentrated 3He
at the temperature T o of the outlet stream of the counterflow heat exchanger. In this
equation it is assumed that the phase boundary between the dilute and the concen-
trated phase is isothermal and that its temperature T m is equal to that of the dilute
solution in the mixing chamber.Greywall measured the specific heat of pure 3He [11]. At temperatures below
50 mK it can be approximated by the expression
Cc(T ) = a · T = 22 · T J
mol · K. (2)
The enthalpy H c(T ) of pure 3He can be calculated by integration and amounts to
H c(T ) = H c(0) + T 0
a · T
· dT
= H c(0) +
a
2 · T 2
. (3)
The specific heat of 3He in dilute solution has been measured by Anderson et al.
[12] at a concentration of 5 percent. Their experimental data can be extrapolated to
6.48 percent (which is the concentration of 3He in the dilute phase inside the mixing
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Fig. 1 (Color online) The CERN Cryolab dilution refrigerator was designed by the authors using the
thermodynamic considerations outlined in this work. This system is of “classical” design, with an inner
vacuum chamber (IVC) immersed in a bath of 4 He at atmospheric pressure. The IVC, which is normally
attached to the 4 K flange, is not shown in the photo above. The 3He is condensed in a continuouslyoperating 1 K pot. Attainable flow rates range from 40 µmol/s to 10 mmol/s, which allows for a thorough
characterization of the system’s performance
chamber of a dilution fridge operating at low temperatures), and can then be described
by the expression
Cd (T ) = b · T = 106 · T J
mol · K. (4)
This number is given per mole of 3He, not per mole of total solution. The chemical
potentials µ(T ) = H ( T ) − T · S(T) of the two phases in the mixing chamber must
be equal. It is thus possible to write
H l (T ) − T ·
T 0
Cd (T )
T · dT = H c(T ) − T ·
T 0
Cc(T )
T · dT , (5)
which can then be used to calculate the enthalpy H l (T ) of 3He in dilute solution
along the solubility line:
H l (T ) = H c(0) +a
2· T 2 − a · T 2 + b · T 2,
(6)
H l (T ) = H c(0) +
b −
a
2
· T 2 = H c(0) + 95 · T
2 J
mol · K2.
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Slightly different values for a and b have been used throughout the years by different
authors; the discrepancies amount to not more than a few percent, however, and are
not important for the theoretical treatment presented in this work.
It must be stressed that H l is the enthalpy of 3He in saturated dilute solution,
which is found in the mixing chamber of a dilution refrigerator. In the heat exchanger,the dilute solution is not saturated. The osmotic enthalpy H d must be used to describe
the enthalpy in the dilute stream of the heat exchanger. This will be detailed later in
this work.
Equations (1), (3) and (6) yield the following formula for the cooling power of a
dilution refrigerator, first put forward by Radebaugh [3]:
Q̇m = ṅ3 · (H l (T m) − H c(T o)) = ṅ3 ·
b −
a
2
· T 2m −
a
2· T 2o
. (7)
Equation (7) can be used to make two simple and well-known statements on dilu-tion refrigerators. First, the available cooling power Q̇m(T m) goes to zero when the
temperature T o of the concentrated stream is
T o =
2b − a
a· T m = 2.94 · T m. (8)
Second, in the theoretical case of T o = T m, which can only be achieved in an hypo-
thetical dilution refrigerator with an infinite heat exchanger surface area, the cooling
power equals to
Q̇m = ṅ3 · (H l (T m) − H c(T m)) = (b − a) · ṅ3 · T 2
m = 84 · ṅ3 · T 2
m
J
mol · K2. (9)
This equation is widely known and often believed to describe the cooling power of
a dilution refrigerator, while in reality it only describes the cooling power of the
actual process of 3He atoms crossing an isothermal phase boundary into dilute so-
lution. The assumption that the surface area of the heat exchanger is infinite is not
realistic. Equation (9) fails to account for the heat transported to the mixing chamber
by the concentrated stream at its temperature T o > T m. This temperature is greatlyinfluenced by the properties of a dilution refrigerator’s heat exchanger.
Also, it is incorrect to assume that a dilution refrigerator yields the highest cooling
power when the temperature of the incoming 3He is as low as possible or as close to
the mixing chamber temperature as possible. It will be shown that, if 3 He is circulated
at a higher rate, the positive effect of more atoms crossing the phase boundary in the
mixing chamber can outweigh the negative effect of a higher heat load to the mixing
chamber due to the increased temperature of the concentrated stream.
2 Maximum Cooling Power and Optimum Flow
Variational calculus can be used to derive very general formulas for the optimum per-
formance of a dilution refrigerator equipped with an ideal continuous countercurrent
heat exchanger. These formulas are derived in the following, and then simplified so
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that analytical expressions are obtained in closed form, using approximations that are
carefully explained and justified. The treatment follows Niinikoski’s original work
[4].
The maximum of the cooling power of (1) is found by requiring that its derivative
with respect to ṅ3 be zero, while keeping the mixing chamber temperature T m andthe still temperature T S constant, i.e.
d Q̇m(T m)
d ṅ3
T m,T S
= 0. (10)
The optimum 3He flow ṅopt, which maximizes the cooling power, is immediately
obtained by applying this to (1) and solving for ṅ3:
ṅopt =H l (T m) − H c(T o)
Cc(T o) · (dT od ṅ3
)T m,T s, (11)
where Cc(T o) is the specific heat of the concentrated stream fluid at the temperature
T o of the concentrated stream outlet of the heat exchanger. This equation can be put
to explicit form by finding the analytical expressions of T o and of its derivative with
respect to ṅ3. These depend on the performance of the heat exchanger and require the
solution of the differential equation describing the heat transfer between the concen-
trated and dilute stream, axial heat conduction and flow friction in this device, which
will be discussed below.
2.1 Heat Transfer, Axial Heat Conduction and Viscous Heating in the Heat
Exchanger
In a continuous countercurrent heat exchanger, the concentrated 3 He flowing into the
mixing chamber is cooled by transferring heat to the colder 3He which is driven in
the dilute stream towards the still by the osmotic pressure gradient. By denoting the
longitudinal coordinate of the heat exchanger by z, with z = 0 in the mixing chamber
and z = L in the still, and by introducing σ (z), the amount of effective heat exchange
surface between 0 and z, the following equation relates the thermal gradient in the
concentrated stream with the fluid and heat exchanger parameters at any position z:
dT c(z)
dz=
α(T c, T d )
ṅ3Cc(T c)
dσ(z)
dz heat exchange
−Ac(z)
ṅ3Cc(T c)
κc(T c)
d 2T c(z)
dz2 +
dκc(T c)
dT c
dT c(z)
dz
2
axial heat conduction
−ηc(T c)V
2c ṅ3
Cc(T c)
dZc(z)
dz frictional heating
. (12)
The subscripts c and d refer to the fluid properties in the concentrated and dilute
streams, respectively. The term α(T c, T d ) is the coefficient of heat transfer between
the two streams. V c, κc(T c) and ηc(T c) are the molar volume, the heat conductivity
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and dynamic viscosity of the concentrated fluid; Ac(z) is the cross sectional area
available for the flow of the concentrated fluid, and Zc(z) is the flow impedance of
the concentrated flow channel.
Equation (12) was obtained by requiring energy balance in each infinitesimal vol-
ume Ac(z) · dz of the concentrated stream. The three terms on the right side stand forthe heat exchange with the dilute stream, axial conduction in the concentrated liquid
and heating of the concentrated stream due to flow friction.
The heat exchange term in (12) should be large, and the terms describing the axial
heat conduction and frictional heating in the concentrated stream, which always have
a negative sign, should be as small as possible. For good heat exchanger designs, axial
conduction in the concentrated and dilute stream, conduction along the mechanical
structures and viscous heating can typically be neglected entirely. Equation (12) can
then be truncated to
dT c(z)
dz=
α(T c, T d )
ṅ3Cc(T c)
dσ(z)
dz. (13)
This equation can be integrated from the outlet temperature T c(0) = T o at z = 0 to the
still temperature T c(L) = T Still at z = L, in order to obtain the relationship between
T o and ṅ3.
2.2 Energy Balance in the Heat Exchanger
In steady state, the relationship T d (z) = f (T c(z)) is unique at a given flow rate, whichfacilitates the analytical integration of (13). Niinikoski used the energy balance at an
arbitrary position z of the heat exchanger to obtain a relation between T c, T d and ṅ3:
H c(T c(z)) +Q̇
ṅ3= H d (T d (z)). (14)
This equation can be obtained by balancing the flow of energy through an imagi-
nary cut through the heat exchanger. Again, axial conduction and viscous heating
are neglected. The approach used to obtain (14) is similar to the one used to obtain(1), where the cut was performed at z = 0. Niinikoski used the data of Radebaugh’s
compilation [3] to numerically solve (14).
Frossati and Takano obtained a relation between T c and T d at any position z of the
heat exchanger by requiring that the heat absorbed by the dilute stream is equal to the
heat given off by the concentrated stream,
ṅ3 · dH c(T c(z)) = ṅ3 · dH d (T d (z),µ4(xm, T m)). (15)
It is noted that the osmotic enthalpy H d is used rather than the enthalpy H l on the
solubility line, since the 3He/ 4He mixture in the dilute stream is not saturated. The
osmotic enthalpy H d depends on the temperature and the chemical potential µ4 of
the 4He component in the dilute solution. The chemical potential µ4 can be assumed
to be constant throughout the system. It is determined by the concentration xm and
temperature T m in the mixing chamber.
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In a first step, the osmotic enthalpy can be approximated by the enthalpy on the
solubility line, and (15) can be written as
ṅ3 · Cc(T c(z)) · dT C = ṅ3 · Cd (T d (z)) · dT d (16)
or
a · T 2c = b · T 2
d (17)
to obtain an estimate for T d /T c. This approach has first been put forward by Frossati
[5]. It must be stressed, however, that this is only an approximation. If the enthalpy
of the dilute stream H d was equal to the enthalpy H l of the 3He/ 4He mixture on
the solubility line, the cooling power of the dilution refrigerator would be zero. In
addition, (16) misses the term Q̇/ṅ3 in the energy balance equation (14), which would
be the constant of integration if (15) were integrated. This error is relatively small,however.
Below 50 mK, the error in T d /T c is smaller than a few percent and it is con-
cluded that T d /T c = ρT ≈ 0.46 in the heat exchanger. For optimum flow conditions,
Niinikoski [4] numerically determined T d /T c = ρT ≈ 0.5 using osmotic enthalpy
data for the liquid in the dilute stream. The relation T d /T c is accurate enough to sim-
plify the heat transfer coefficient to α(T c), which then only depends on T c. It cannot
be used to determine the temperature of the concentrated 3He entering the mixing
chamber, however. It cannot be used at arbitrary flow rates, either.
2.3 Heat Transfer Between the Concentrated and Dilute Stream by Kapitza
Conductance
Large transverse thermal conductance between the concentrated and dilute streams
is commonly achieved by using sintered sponges with a large surface area. These
sponges are made from materials of high thermal conductivity (typically, silver or
copper are used). Axial conduction can be reduced by dividing the sintered part of
the heat exchanger into several subsections, and joining these to each other in series
with a material of low thermal conductivity. Axial conduction will thus be neglectedin this analytical treatment of the heat exchanger.
The transverse heat transfer parameter is determined by the thermal conductivities
of the helium fluids in the pores of the sponge, the separating wall materials and,
in particular, by the Kapitza conductances between the helium fluids and the solid
wall. However, if the wall surface is extended by fine sintered powder, the Kapitza
resistance may become smaller than the other thermal resistances; this typically is
the case for high temperatures T > 100 mK. At lower temperatures, the Kapitza
conductance often is the limiting factor, and the heat Q̇ transferred between the
concentrated and dilute streams obeys the relation
Q̇ = σ cβc
T nc − T n
W
= σ d βd
T nW − T
nd
, (18)
where βc(d) are the Kapitza conductances at the interfaces of the sintered powder and
the concentrated (dilute) fluids, T W the temperature of the separating wall, and n ≈ 4
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is approximately constant in a wide range of temperatures.1 The temperature of the
separating wall T W can be eliminated from (18) to get
Q̇ = σ c
βc
T n
c
1 − ρnT
1 + ρβ= σ
cST n
c , (19)
where ρT = T d /T c , as previously defined, and
ρβ =σ cβc
σ d βd . (20)
As n ≈ 4, the uncertainty in T d /T c has only a very small influence on the result. For
simplicity, the parameter S is introduced, which can be understood as a corrected
Kapitza conductance:
S = βc1 − ρnT 1 + ρβ
. (21)
The ratio ρβ of the Kapitza conductances per unit length of the heat exchanger be-
tween the concentrated and dilute streams is constant provided that the ratio of the
sintered contact surfaces is constant. As the variables on the right side of (20) are
known for a given heat exchanger design, ρβ can be calculated.2
It must be stressed that (19) is a good approximation of (18) only when a dilutionrefrigerator is operated at or close to the maximum power, but not when the 3He flow
rate is far from optimum. Small variations in ρT and ρβ do not produce a substantial
change in S , which therefore remains a constant within an accuracy that is at least as
good as the power law approximation of (18) for the Kapitza resistance itself.
In conclusion, T d can be eliminated from the expression of α(T c, T d ) and one may
write
α(T c, T d ) = αc(T c) = S · T n
c , (22)
where n ∼= 4 as before. It must be stressed that the approximate thermal resistance
described by (22) depends on the temperature of the concentrated stream alone. The
error of this approximation is small since the temperature of the concentrated stream
is about twice as high than that of the dilute stream at any point z along the heat
exchanger at optimum flow, which is why the Kapitza conductance of (18) depends
very weakly on the temperature of the dilute stream, and why neglecting the term
Q̇/ṅ3 in (14) is not fatal.
1In the literature, the thermal boundary resistance is often described using the value RK , where Q̇ =
T/RK . The thermal boundary resistance RK is then proportional to T −3, and one can write β = 1/(4 ·
RK · σ · T 3).
2Many authors assume equal surface areas and heat transfer coefficients in the concentrated and dilute
stream, i.e. ρβ = 1. In that case, S would amount to S = (1 − (a/b)2/8RK · σ · T
3).
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2.4 Maximum Cooling Power and Optimum Flow Rate
Finally, the truncated equation (13) can be integrated along the concentrated stream
(i.e. from its cold end at z = 0 and T o to its warm end at z = L and T S ) to get
σ c = ṅ3 ·
T S T o
Cc(T c)
αc(T c)dT c, (23)
where σ c is the integrated effective surface area of the concentrated stream. The de-
nominator of (11) is now obtained by differentiating with respect to ṅ3 while keeping
all other variables but T o constant and by rearranging the terms to get
σ c
ṅ3= ṅ3
Cc(T o)
αc(T o)dT o
d ṅ3T m,T s . (24)This gives
Cc(T o)
dT o
d ṅ3
T m,T s
=σ cαc(T o)
ṅ23
, (25)
which can be inserted in (11) to yield the optimum flow rate of 3He,
ṅopt =σ cαc(T o)
H l (T m) − H c(T o), (26)
and, by inserting this into (1), the maximum cooling power
Q̇max(T m) = σ cαc(T o). (27)
The results of (26) and (27) are valid for all T m at optimum flow. The outlet tem-
perature T o of the heat exchanger can be evaluated with a precision better than the
approximation T o = T m/ρT used above for the heat exchanger. This can be achieved
by combining (23) and (26) to get
αc(T o)
T sT 0
Cc(T c)
αc(T c)dT c = H l (T m) − H c(T o), (28)
whose solution yields the ratio T o/T m as a function of T m—as was obtained in [4].
The dependence of this ratio on the mixing chamber temperature T m is universal for
a given temperature dependence of the thermal resistance αc(T c). The exact solu-
tion of this equation shows that the ratio T o/T m approaches an asymptotic value at
temperatures well below 50 mK, where the specific heat and the enthalpies can be ap-
proximated by (2), (3) and (6), and the heat transfer parameter αc by (22). Inserting
these into (28) yields
a · T no ·
T 2−nS − T
2−no
2 − n
=
b −
a
2
· T 2m −
a
2· T 2o . (29)
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With n = 4 and T −2S T −2
o the following relation for T o/T m is obtained:
T o
T m∼=
2b − a
2a∼= 2.1, (30)
which is fairly close to the value obtained in (17), but more accurate. It must be
stressed that this equation is only valid at optimum flow conditions, i.e. when the
cooling power reaches a maximum for a given mixing chamber temperature. The
asymptotic expressions for the maximum cooling power and the corresponding opti-
mum 3He flow rate can now be written:
Q̇max =
2b − a
2a
2σ cST
4m
∼= 18.6 · σ c · S · T 4
m, (31)
ṅopt =2b − a
a2 σ cST 2m ∼= 0.39 · σ c · S · T 2m
K2 mol
J . (32)
At temperatures below 50 mK the maximum cooling power is therefore proportional
to T 4m and to the effective total surface area of the heat exchanger; this maximum
power is obtained at an optimum flow that is proportional to T 2m and to the effective
surface area.
3 The Mixing Chamber Temperature at Arbitrary Flow Rates
Frossati and coworkers [5] and [6], who are known for designing some of the world’s
most powerful dilution refrigerators, attempted to model a dilution refrigerator such
that performance predictions can also be made at 3He flow rates away from the op-
timum flow rate given by (32). Such flow conditions may occur in practice when the
heat leak to the still limits the minimum 3He flow rate, or when a high optimum 3He
flow rate cannot be supported by the pumps, the precooling system or the condenser.
The latter condition may also occur when compressible flow effects occur in the 3He
pump line. This can be the case when too small an orifice is used to reduce superfluid
film flow in the vapor outlet of the still.
Frossati’s method consists of determining the asymptotic dependence of the mix-ing chamber temperature T m at
3He flow rates far below and far above the optimum,
and of then combining the two formulas to obtain an intermediate behavior for T m.
In the following, the work of Frossati and his coworkers is reviewed using this
work’s notation for the parameters describing the enthalpies and the Kapitza conduc-
tance, which are slightly different from those used in their original work. Their results
can then easily be compared with the maximum cooling power and the optimum flow
given by (31) and (32).
At a flow far below optimum, the heat exchanger outlet temperature T o will be
very close to T m, so that T o can be replaced with T m in (1), in exactly the sameway which was used to obtain (9). At a flow very much below optimum, the mixing
chamber temperature T m can be calculated by
T 2m =Q̇m
ṅ3(b − a). (33)
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On the other hand, at a flow much higher than the optimum, the heat load is much
lower than the rate at which the concentrated stream carries enthalpy into the mixing
chamber. Q̇m can then be neglected in (1), which can be solved for T m in the same
way which was used to obtain (8):
T 2m =a
2b − aT 2o . (34)
Under these conditions the flow dependence of T o is obtained by directly integrating
(23)
σ c = ṅ3 ·
T S T o
a · T c
ST 4cdT c =
ṅ3 · a
S ·
T S T o
dT c
T 3c=
ṅ3 · a
2 · S ·
1
T 2o−
1
T 2s
, (35)
where the second term inside the square brackets can be neglected at very low tem-
peratures of the mixing chamber. By inserting relation (34) one finally obtains
T 2m =a2 · ṅ3
(4b − 2a) · σ c · S . (36)
There are now two expressions for T 2m, of which (33) is valid at flow rates well be-
low the optimum, and (36) at flows well above the optimum. Frossati and coworkers
simply added these two expressions to obtain their result
T 2m =a2 · ṅ
3(4b − 2a) · σ c · S
+Q̇
mṅ3(b − a)
(37)
which converges correctly at flow rates far above and far below the optimum. In the
important range close to optimum flow rates, however, the characteristic of the refrig-
erator’s heat exchanger is not correctly taken into account and the results yielded by
(37) are not valid. This is due to the fact that the addition performed to obtain (37)
cannot be derived from any basic thermodynamic law.
In the following, the optimum flow and maximum cooling power that can be ob-
tained from (37) are compared with (31) and (32). The partial derivative of (37) is
zero at minimum temperature for a given constant power:
∂(T 2m)
∂ṅ3=
a2
(4b − 2a) · σ c · S −
Q̇m
ṅ23(b − a)= 0, (38)
which yields directly the relation for the optimum flow rate and maximum cooling
power:
ṅopt = Q̇maxσ cS ·4b − 2a
a2 · (b − a). (39)
By inserting this into (37), one can solve for the optimum flow and maximum power
as a function of mixing chamber temperature:
Q̇max = σ c · S · T 4
m ·(4b − 2a) · (b − a)
4a2 (40)
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and
ṅopt = σ c · S · T 2
m ·2b − a
a2 . (41)
It is noted that the above relation for the optimum flow rate is identical to (32). The
relation for the maximum cooling power, however, is different from the one calcu-
lated in (31), although it shows the same dependence on the heat exchanger surface
area, Kapitza conductance and mixing chamber temperature. The difference between
(31) and (40) is due to the incorrect addition in (37). The maximum cooling power
yielded by (40) is not correct and amounts to only approximately 88 percent of the
actual cooling power. The minimum mixing chamber temperature yielded by (40) is
3 percent higher than the actual temperature.
In 1992, Frossati improved his model by combining Radebaugh’s equation (7)
with (34) and (36) to obtain an expression for the mixing chamber temperature [7].
His new result for the mixing chamber temperature of a dilution fridge thus was
T 2m =a2 · ṅ3
(4b − 2a) · σ c · S +
Q̇m
ṅ3(b − a/2). (42)
Performing the same steps as above leads to the following equation for the maximum
cooling power:
Q̇max = σ c · S · T 4
m ·2b − a
2a 2
, (43)
which is identical to (31). The relation for the optimum flow rate remains the same
as in (32) and (41). A comparison between Niinikoski’s model and Frossati’s original
and revised models is shown in Fig. 2.
In his 1994 paper [8], Takano stated that, due to a difficulty with the boundary
conditions used in the treatment of the heat exchanger, (42) was only valid close to
the base temperature of a dilution refrigerator. More precisely, (42) is only valid at
optimum flow conditions,3 and a reproduction of Niinikoski’s original result.
It is interesting that (42) yields the same result for optimum conditions as (31)
despite of calculating the concentrated stream’s outlet temperature T o differently.
Equation (30) gives the relation between the outlet temperature T o and the mixingchamber temperature T m at the optimum flow rate given by (32). These equations are
part of Niinikoski’s derivation. Frossati used (34) for the relation between T o and T m.
This equation is valid at the flow rate of (36), which is higher than the optimal flow
rate (it is, in fact, the flow rate which would, for a given heat exchanger, lead to a
given mixing chamber temperature in the absence of an external heat load). In both
cases, the following relation is obtained for the outlet temperature T o:
T 2o =a
2
·ṅ3
σ c · S
. (44)
3The minimum temperature achieved at optimum flow conditions must not be confused with a refriger-
ator’s “base temperature”. The base temperature is the lowest temperature which can be achieved with
a given dilution refrigerator. It does not necessarily occur at the optimum flow rate, which may not be
attainable due to too high a heat load on the still, for instance.
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Fig. 2 (Color online) Comparison between (31) and (32) for the optimum flow results (solid dot ), Frossati
et al.’s initial model (equation (37), dotted line) and Frossati’s revised model (equation (42), solid line).
The calculations were performed for a fictional dilution refrigerator with surface areas σ c = σ d = 0.1 m2,
Kapitza conductances βc = βd = 15 Wm−2K−4 and a heat load of 100 nW. For a given heat load, the
minimum temperature calculated with (37) (dotted line) is shifted to flow rates slightly higher than theactual optimum as the calculated minimum temperature is higher than the actual one
This agreement between the two models is a coincidence, which occurs due to the fact
that, at optimum flow, the heat load on the mixing chamber Q̇m is exactly equal to the
enthalpy ṅ3 · H c(T o) carried into the mixing chamber by the concentrated stream. The
flow rate calculated from (36) is thus exactly twice as high as the optimum flow rate.
On the other hand, the ratio T 2o / T 2
m yielded by (34) at that flow rate is also exactly two
times higher than the ratio T 2o
/ T 2
m at optimum flow (this is evident as the enthalpy
carried into the mixing chamber by the concentrated stream is proportional to ṅ3 in
the absence of an external heat load Q̇m, which is assumed in Frossati’s treatment
of the heat exchanger). These two inaccuracies in Frossati’s model cancel each other
out to yield the correct outlet temperature T o at optimum flow.
In a rough approximation, (30) can be used to relate the outlet temperature T o to
the temperature T i at which liquid enters the dilute stream:
T 2i =a2
(2b − a)
·ṅ3
σ c · S
. (45)
It must again be pointed out that (30) is only accurate at optimum flow conditions.
The same is thus true for (45).
The temperatures T o and T i obtained with (44) and (45) are shown in Fig. 3 along
with the mixing chamber temperature obtained with (42). As Takano pointed out in
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Fig. 3 (Color online) The mixing chamber temperature (equation (42), solid line), the outlet temperature
of the concentrated stream (equation (44), dashed line) and the temperature of the liquid entering the dilute
stream (equation (45), dot-dashed line), all calculated with Frossati’s revised model. The calculations were
performed for the same fictional dilution refrigerator of Fig. 2
[8], T o and T i drop below the mixing chamber temperature at low flow rates, which
is clearly unphysical. In addition, T i must always be equal to T m.
Takano [8] also derived an equation for the cooling power as a function of the
mixing chamber temperature and the 3He flow rate, which is valid at non-optimum
conditions:
Q̇ =b
2− a
2
· ṅ3 · T 2m − a · b2
· ṅ23 · 1 + ρβ
4 · σ c · βc· ln (a + b) · [(b − a) · T 2m − Q̇/ṅ3]
(b − a) · (b · T 2m − Q̇/ṅ3)
.
(46)
Due to its implicit character, this equation cannot be solved for T m analytically. The
numerical solution is slightly complicated by the fact that two valid results are yielded
when T m is calculated for a given flow rate and heat load. This is illustrated in the
graphical solution of (46) shown in Fig. 4.
Figure 5 shows the numerically obtained solution of (46) and compares it with
Frossati’s revised model of (42).
The differences between Niinikoski’s and Frossati’s analytical models and
Takano’s numerical results are hardly discernible, but exist. They occur due to the
assumption which was made to obtain (22), namely that the heat transfer between
the dilute and the concentrated stream is independent of the temperature of the dilute
stream.
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Fig. 4 (Color online) Equation (46) yields two valid results for the mixing chamber temperature. The
horizontal solid line is the left side of (46), the dotted line is the right side of (46). The two intersections
are valid solutions of (46), and it is difficult to determine the correct one from intuition. The calculations
were performed assuming optimum flow and a heat load of 100 nW for the fictional dilution refrigerator
of Fig. 2
Fig. 5 (Color online) The mixing chamber temperature yielded by Niinikoski’s model (equations (31) and
(32), !), Frossati’s revised model (equation (42), solid line) and Takano’s model (equation (46), dotted
line). The minimum temperature and optimum flow yielded by Takano’s model are marked with ×. The
calculations were performed for the fictional dilution refrigerator of Fig. 2
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Fig. 6 (Color online) A schematic of the nodal model used to simulate a dilution refrigerator. The white
nodes represent the dilute and concentrated streams, the white arrows the Kapitza conductors and the black
arrows the
3
He flow through the system. Only seven node pairs are shown to represent the heat exchanger,while 1000 pairs have been used for the actual simulation
4 A New Numerical Model for Arbitrary Flow Rates
The performance of a dilution refrigerator at arbitrary flow rates can thus only be ex-
actly calculated with numerical methods. Another such method, using a nodal model-
ing technique, has been developed at CERN in the framework of a project which aims
at developing powerful dilution refrigerators for future generations of cryogenic dark matter detector arrays [13]. In addition to being capable of correctly evaluating the
performance at both optimum and non-optimum flow conditions, transient behavior
can be modeled. Also, non-ideal effects such as axial conduction, viscous heating, the
effects of non-continuous heat exchangers, and thermal penetration effects occurring
in sintered sponges can be correctly taken into account. Since the fluid properties of
the helium liquids do not have to be expressed by rather simple analytic functions,
the numerical method is valid at temperatures as high as several hundred millikelvin.
In this section, a nodal model for an ideal dilution refrigerator with a perfect coun-
tercurrent heat exchanger is described. The heat capacities given by (2) and (4) areused to facilitate comparison with the previous sections of this work. For the same
reason, the difference between the enthalpy on the solubility line and the osmotic
enthalpy of dilute solution is neglected.
In the nodal model, the dilution refrigerator is modeled as a number of discrete
nodes which are linked by conductors. A solution is then obtained by solving for
thermal equilibrium, i.e. by determining the temperatures of all nodes such that the
heat flow into each node is equal to the heat flow out of each node.
The mixing chamber is represented by a node to which the external heat load Q̇mand the cooling power are applied. The cooling process is modeled as a negative heat
load with the magnitude given by (9). The still is represented by a node of constant
temperature T S = 700 mK. The heat exchanger is modeled by a group of nodes repre-
senting the concentrated stream and a group of nodes representing the dilute stream.
Each node C in the concentrated stream is connected to a node D in the dilute stream
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Fig. 7 (Color online) Comparison between Takano’s numeric results (solid line) and the nodal method
(solid dots). The calculations were again performed for the fictional dilution refrigerator with surface areas
σ c = σ d = 0.1 m2, Kapitza conductances βc = βd = 15 W m
−2 K−4 and a heat load of 100 nW
with a conductor element along which the heat flow amounts to
Q̇Kapitza =σ c · βc
1 + ρβ·
T 4c − T 4
d
. (47)
This accounts for the Kapitza heat transfer between the concentrated and dilute
stream. In (47), σ c stands for the surface area of the heat exchanger in the section of
the concentrated stream which is represented by node C, βc the Kapitza conductances
in the concentrated stream, and ρβ the characteristic number defined in (20).Fluid flow is modeled by connecting the nodes of each stream in series with linear
elements. The heat flow along these linear conductors between two adjacent nodes n
and n + 1 is given by
Q̇Flow = ṅ3 · C(T n) · (T n − T n+1), (48)
where C is the heat capacity of either pure 3He (for the concentrated stream) or 3He
in 4He (for the dilute stream). Heat can only be transported in the direction of the
fluid flow. This approach is equivalent to solving (13) numerically.
In Fig. 7, the results yielded by Takano’s model and the nodal method described
in this section are compared with each other. As in the earlier figures, the data is
for a hypothetical dilution refrigerator with βc = βd = 15 W/m2 K4 and σ c = σ d =
0.1 m2 running with an external heat load of 100 nW. The two models are in excellent
agreement.
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Fig. 8 (Color online) Comparison between the cooling power of a dilution refrigerator with an ideal con-
tinuous countercurrent heat exchanger (solid line), and the same refrigerator with two step heat exchangers
(dashed line)
while the 3He component is considered to be in gaseous form at a pressure which
is equal to the osmotic pressure. The molar volume of 3He in the mixture can be
assumed to be constant (V 3 = 4.15 · 10−4 m3 /mol) and the viscosity obeys the law
η(T ) =η0
T 2, (51)
where η0 = 5 · 10−8 Pa K2 s [18]. It has to be noted that the effects of viscous
heating and conduction in the helium liquids are usually negligible.
For the fictional dilution refrigerator of this work, with its 1 m long heat exchanger
with 3 mm2 flow cross sections, the mixing chamber temperature does not rise by
more than 58 µK if viscous heating is taken into account. Viscous heating can be
made negligible if Niinikoski’s design rules [4] are followed.
• The assumption, that the countercurrent heat exchanger in a dilution refrigerator
is ideally continuous, can yield to large errors for some units. It has come to the
authors’ attention that at least one popular commercially available dilution refrig-
erator has only two sintered step heat exchanger elements. In this case, it is im-
possible to treat the countercurrent heat exchanger as ideally continuous. A correct
treatment is only possible with the nodal model.
In Fig. 8, the discrepancy between the ideally continuous model and the model
with two step heat exchangers is illustrated. It was assumed that 10 percent of
the total heat exchanger surface area are in a tube-in-tube countercurrent heat ex-
changer preceding the sintered heat exchangers. The remaining surface was distrib-
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uted equally over two isothermal step heat exchanger elements. The degradation
over an ideal continuous heat exchanger is considerable.
• If sintered heat exchangers are used, the thermal penetration depth of the helium
liquids into the sponge has to be taken into account. This effect, which is due to the
limited thermal conductivity of the helium liquids filling the pores of the sponge,reduces the effective surface area, and σ c and σ d have to be adjusted accordingly.
This effect has been studied in detail by Cousins et al. and has been reported in [ 19]
and [20]. While in theory it is possible to integrate this effect into a nodal model,
such attempts performed at CERN have not been exceedingly successful [13].
6 Conclusions
The theory of the thermodynamic processes underlying dilution refrigeration is well
understood. The maximum cooling power of an ideal dilution refrigerator is propor-
tional to T 4m, and can be evaluated analytically with reasonable accuracy.
In contrast to general opinion, it is not possible to evaluate the exact cooling power
at an arbitrary flow rate with analytical means, and numeric methods have to be used.
A new numeric method, which offers the possibility to account for non-ideal effects,
such as the limited number of step heat exchangers, has been presented in this paper.
Experimental uncertainties can easily cause errors in the range or even larger than
the deviations caused by these non-ideal effects, most prominently by the temperature
dependent efficiency of sintered heat exchangers.
Acknowledgements Patrick Wikus wishes to thank Prof. Harald Weber of the Institute of Atomic and
Subatomic Physics in Vienna, and Prof. Chris Fabjan, Gerhard Burghart and the staff of the Central Cryo-
genic Laboratory at CERN for their support in this work.
This research project on dilution refrigeration was partly carried out in the framework of the Austrian
Doctoral Students Program at CERN. Funding has been provided by the Austrian Federal Ministry for
Education, Science and Culture.
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