Paper presented at the Cryogenic Engineering Conf., Columbus, Ohio, July
1995

To be published in: Advances in Cryogenic Engineering, vol. 41

**THEORETICAL ANALYSIS OF A PULSE TUBE**

**REGENERATOR**

> Pat R. Roach1, Ali Kashani2 and J. M. Lee1
>
> 1NASA Ames Research Center
>
> Moffett Field, CA 94037-1000
>
> 2Atlas Scientific
>
> Sunnyvale, CA 94086

**Abstract**

A theoretical analysis of the behavior of a typical pulse tube
regenerator has been carried out. Assuming simple sinusoidal
oscillations, the static and oscillatory pressures, velocities and
temperatures have been determined for a model that includes a
compressible gas and imperfect thermal contact between the gas and the
regenerator matrix. For realistic material parameters, the analysis
reveals that the pressure and velocity oscillations are largely
independent of details of the thermal contact between the gas and the
solid matrix. Only the temperature oscillations depend on this contact.
Suggestions for optimizing the design of a regenerator are given.

**Conservation Equations**

The conservation equations for an element of gas undergoing
three-dimensional flow and exchanging heat with its surroundings,
ignoring gravitational forces, can be written as follows1 (we adopt the
notation that variables with an asterisk are the normal unscaled
variables; later we will introduce scaled, dimensionless variables,
which will not have an asterisk):

![](media/image1.pct){width="6.041666666666667in"
height="1.4166666666666667in"}

where **v\*** is the vector velocity, **τ\*** is the stress tensor,
**q\*** is heat flow, and U\* is the internal energy density.

In the case of one-dimensional flow in the z-direction, eqs. (1), (2)
and (3) become (using eq. (1\') to arrive at the form of eqs. (2\') and
(3\')), in cylindrical coordinates:

![](media/image2.pct){width="6.25in"
height="3.0416666666666665in"}![](media/image3.pct){width="6.25in"
height="3.0277777777777777in"}where Kp is the Darcy permeability.

To convert the variables to dimensionless form we use the following
scaling parameters: P0\*, average pressure in the system, T0\*,
temperature at hot end of regenerator, ρ0\*, density at hot end of
regenerator, q0\*, gas velocity at hot end of regenerator, L0\*, length
of regenerator and τ\*, period of oscillation. These scaling parameters
can be combined into dimensionless parameters as follows:

![](media/image4.pct){width="6.25in" height="0.5972222222222222in"}These
parameters are convenient when expressing the previous equations in
dimensionless form:

![](media/image5.pct){width="6.25in" height="1.5416666666666667in"}

![](media/image6.pct){width="6.041666666666667in"
height="0.8333333333333334in"}

![](media/image7.pct){width="6.208333333333333in"
height="0.9722222222222222in"}![](media/image8.pct){width="6.263888888888889in"
height="2.6944444444444446in"}

**SOLUTION OF EQUATIONS**

These equations will be solved in a perturbation treatment that is valid
when the level of oscillation of the gas is small. The perturbation
parameter is ε which is proportional to q0\*, the amplitude of the
velocity oscillation at the hot end of the regenerator. Further, it is
assumed that all the important features of the system operation can be
described with only fundamental frequency terms and that higher harmonic
terms can be neglected. These assumptions lead to the following
expansions for the dimensionless variables in the above equations:

![](media/image9.pct){width="6.125in" height="1.4722222222222223in"}

Subscripts [a]{.underline} refer to average values and subscripts
[d]{.underline} refer to the amplitude of oscillating or dynamic
variables.

When eqs. (12) are substituted into eqs. (6), (7), (8), (10) and (11)
the lowest order equations describing the oscillating-term coefficients
are:

![](media/image10.pct){width="6.125in" height="1.1666666666666667in"}

![](media/image11.pct){width="6.125in"
height="1.1944444444444444in"}Equation (17) is easily solved if the last
term, km∂2Tmd/∂z2, is neglected and if the oscillating matrix
temperature is assumed to be proportional to the gas temperature
oscillations, but with a phase shift. Then eq. 16 becomes (neglecting k
∂2Td/∂z2):

![](media/image12.pct){width="6.125in" height="1.1111111111111112in"}

The variables ρ0, P0 and T0 are determined by lower order equations and
by static boundary conditions. The results are P0 = 1, T0 = 1+ΔT z and
ρ0 = 1/T0, where ΔT is the temperature difference from the hot to the
cold end of the regenerator. Equations (14), (15) and (18) can be used
to express the three oscillating-term coefficients, qd, Td and ρd in
terms of Pd and eq. (13) can use these expressions to arrive at the
differential equation2 for Pd:

![](media/image13.pct){width="6.166666666666667in"
height="1.0972222222222223in"}

With specified boundary conditions, eq. (19) can be solved for Pd by
numerical methods and the result used to find qd, Td and ρd. The
boundary conditions will be provided by the requirements of matching
pressures and mass flows at the two ends of the regenerator\--at the
compressor (hot end) and at the pulse tube (cold end).

There is one problem, however. When average enthalpy flow (time-averaged
over one cycle) in the regenerator is calculated from these oscillating
parameters, one finds:

![](media/image14.pct){width="6.208333333333333in"
height="1.3194444444444444in"} In general this is not constant with z in
the regenerator. This means that there must be some conversion of
enthalpy flux into heat that is conducted axially through the gas and/or
conducted into the matrix and conducted axially through the matrix. For
this to happen there must be a non-linear static temperature profile in
the gas and/or the matrix. But T0 is linear in z and it can be shown2
that neither Ta nor Tma can provide the necessary temperature profile
correction. Therefore, the correction to the linear T0 profile must come
from T2a.

![](media/image15.pct){width="6.25in" height="1.0138888888888888in"}

![](media/image16.pct){width="6.125in" height="0.9722222222222222in"} We
now have the corrected static temperature profile and the lowest order
oscillating terms, Pd, Td, ρd and qd. In addition, we can calculate the
average enthalpy flux and it obeys the second order energy conservation
equations.

**boundary conditions at compressor**

![](media/image17.pct){width="6.208333333333333in" height="1.625in"} For
an isothermal compressor, (essentially the same whether scaled or
unscaled)

![](media/image18.pct){width="6.25in"
height="1.5in"}![](media/image19.pct){width="6.222222222222222in"
height="0.8472222222222222in"}This equation provides one of the boundary
conditions for the solution of eq. (19).

**boundary conditions at Pulse Tube**

![](media/image20.pct){width="6.208333333333333in"
height="1.4166666666666667in"}

![](media/image21.pct){width="6.125in" height="0.8611111111111112in"}

![](media/image22.pct){width="6.25in"
height="0.875in"}![](media/image23.pct){width="6.125in"
height="0.5277777777777778in"}

![](media/image24.pct){width="6.208333333333333in" height="1.75in"}

This equation provides the second boundary condition for the solution of
eq. (19).

At the present time, the solution of eq. (19) is carried out with
temperature-independent values for thermal conductivity, viscosity and
heat transfer coefficient. The effect of a temperature-dependent
viscosity will be incorporated in the next improvement of the
calculation. We are looking into ways to include the temperature
dependence of the thermal conductivity and the temperature and velocity
dependence of the heat transfer coefficient.

If all the dimensions and parameters of the system are inserted into the
equations, the results for cooling power can be calculated. The enthalpy
flow in the pulse tube is the fundamental cooling of the system from
which various heat leaks must be subtracted. These are: axial conduction
in the wall of the pulse tube, axial conduction in the wall and matrix
of the regenerator, and enthalpy flux in the gas at the hot end of the
regenerator. (This last term redistributes itself to become an enthalpy
flux in the gas at the cold end of the regenerator plus an additional
axial conduction through the wall and matrix as discussed above; the sum
of these two contributions doesn\'t change, however, as it passes
through the regenerator.) The orifice opening parameter is then varied
to find a value that maximizes the net cooling power.

**Results from Model**

When typical numbers are used for the various parameters in the above
equations, it is noticed that the two terms in eq. (19) with
denominators (Γ T0 + 1)2 + α2 are very small, approximately 2 orders of
magnitude less than the other terms in the equation. These terms come
from the expression for the oscillating temperature. It would not
distort the solution appreciably to ignore these terms in the
calculation of the pressure. This is plausible since it would not be
expected that the thermal behavior of the system would have a large
effect on the mechanical behavior (pressure and velocity) of the system.
Once the pressure and velocity were found from the simpler approach then
the temperature oscillation could be found from the eq. (18). This is
not the approach we take but it would certainly be reasonable if
circumstances made it advantageous.

![](media/image25.pct){width="6.069444444444445in"
height="4.638888888888889in"}

It is interesting to ask how the performance of a pulse tube depends on
the length-to-diameter ratio of the regenerator. For a constant volume
of the regenerator, there should be an optimum length-to-diameter ratio
since the axial-conduction heat leak will become very large at short
lengths and the pressure drop will become very large at long lengths. We
used our model to calculate some performance data for a typical orifice
pulse tube operating between 300 K and 100 K using helium gas with
γ=1.68. The cooling power we calculate is composed of the enthalpy flow
in the pulse tube section reduced by the enthalpy flow through the
regenerator toward the cold end and reduced by axial heat conduction
through the walls of the pulse tube and regenerator. Figure 1 shows the
results for a regenerator filled with 150-mesh screens. The optimum
length/diameter ratio is 12 for this case.

When the regenerator screens are made finer by going to 250 mesh, the
cooling power is higher at the optimum length/diameter ratio as shown in
fig. 2. Because of the more-restrictive screens, the optimum
length/diameter occurs at 3.6.

![](media/image26.pct){width="6.069444444444445in"
height="4.638888888888889in"}

**References**

1\. R. Byron Bird, Warren E. Stewart and Edwin N. Lightfoot, *Transport
Phenomena,* John Wiley & Sons, New York, (1960).

2\. Pat R. Roach, Ali Kashani and J. M. Lee, *Cryogenics* (to be
published).

3\. Peter J. Storch, Ray Radebaugh and James E. Zimmerman, NIST
Technical Note 1343.