Theoretical Models for the Cooling Power and Temperature of Dilution Refrigerators

8/19/2019 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

mailto:[email protected]:[email protected]


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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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J Low Temp Phys (2010) 158: 901–921 913

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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914 J Low Temp Phys (2010) 158: 901–921

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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J Low Temp Phys (2010) 158: 901–921 915

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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916 J Low Temp Phys (2010) 158: 901–921

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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J Low Temp Phys (2010) 158: 901–921 917

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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J Low Temp Phys (2010) 158: 901–921 919

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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920 J Low Temp Phys (2010) 158: 901–921

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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Theoretical Models for the Cooling Power and Temperature of Dilution Refrigerators