Tom Hofler's thermoacoustic refrigerator was described in detail in his Ph. D. thesis ``Thermoacoustic Refrigerator Design and Performance," UC San Diego, Physics Department (1986). The work was also summarized in the proceedings of the 5th International Cryocoolers Conference, 1988, Monterey CA, p. 93. We use this case to further illustrate capabilities of DELTAE, generating curves similar to Figs. 16 and 17 in Hofler's thesis (Figs. 5 and 6 in the Cryocoolers proceedings).
The apparatus is shown in Fig. III.3:
Figure III.3: Hofler's thermoacoustic refrigerator.
We began with an input file (hofler.in, in the examples directory) whose geometry is that of Hofler's `long' apparatus:
=0.8
TITLE Hofler's 1986 thermoacoustic refrigerator ! Geometry comes from Hofler thesis, pages 28, 64, 68, 115, 130, 133. BEGIN 1.0e6 Pa Mean P 500. Hz Freq. 300. K T-beg 3.0e4 Pa |p|@0 0.0 deg Ph(p)0 5.0e-4 m3/s |U|@0 0.000 deg Ph(U)0 helium Gas ENDCAP driver end 1 1.134e-3 m2 Area SAMEAS 0 Gas ISODUCT room temp duct 2 SAMEAS 1 Area 0.119 m Perim 4.26e-2 m Length SAMEAS 0 Gas HXFRST room temp heat exchanger 3 SAMEAS 1 Area 0.600 GasA/A 6.35e-3 m Length 1.9e-4 m y0 -20.0 W HeatIn 300. K Est-T (I hope this was the experimental value.) SAMEAS 0 Gas STKSLAB Stack 4 SAMEAS 1 Area 0.724 GasA/A 7.85e-2 m Length 1.8e-4 m y0 4.0e-5 m Lplate SAMEAS 0 Gas kapton Solid HXLAST Cold heat exchanger 5 SAMEAS 1 Area 0.67 GasA/A 2.54e-3 m Length 2.55e-4 m y0 3.0 W Heatin 200. K Est-T SAMEAS 0 Gas ISODUCT Cold Duct 6 3.84e-4 m2 Area 0.0694 m Perim 0.167 m Length SAMEAS 0 Gas ISOCONE 7 SAMEAS 6 Initial Area SAMEAS 6 In Perim 6.68e-2 m Length 1.16e-3 m2 Final area 0.121 m Final perim SAMEAS 0 COMPLIANCE end bulb 8 0.049 m2 Area 1.06e-3 m3 Volume SAMEAS 0 Gas HARDEND 0.000 R(Zin) 0.000 I(Zin) SAMEAS 0 Gas type
Note the use of segment type STKSLab to model the parallel-plate stack geometry, and the use of segment types ISOCOne and COMPLiance to model parts of the cold portion of the resonator.
Executing DELTAE and choosing this input file, we used (C)lear|set to ask for default targets,
responding with `h' because we now have a heat pump, and examine the vector status summary:No vectors defined...do you want enable a default set of targets&guesses for this model? (y/n) y Is this a prime-mover or a heat pump(p|h)? h
Iteration Vectors Summary: GUESS 0b 0c 0f name BEGIN:Freq. BEGIN:T-beg BEGIN:|U|@0 value 5.00E+02 3.00E+02 5.00E-04 units Hz K m^3/s TARGET 3f 9a 9b name HXFRS:Est-T HARDE:R(1/Z HARDE:I(1/Z units K value 3.00E+02 0.00 0.00 Potential TARGETS still available: Addr Seg:Par-Type Current Value 5e HXLAST:HeatIn= 3.000 W 5f HXLAST:Est-T = 200.0 K
This time we are not satisfied with DELTAE's default choice of elements of this table. We would like to generate a curve like Hofler's Fig. 16. To show off DELTAE's ability to handle more dimensions in its search algorithm, and to get a direct grip on the independent variable in Hofler's figure, we made the refrigeration power a target, adding the room temperature waste heat to the guess vector. After making these changes, the vector summary looks like
Iteration Vectors Summary: GUESS 0b 0c 0f 3e name BEGIN:Freq. BEGIN:T-beg BEGIN:|U|@0 HXFRS:HeatI value 5.00E+02 3.00E+02 5.00E-04 -20.0 units Hz K m^3/s W TARGET 3f 5e 9a 9b name HXFRS:Est-T HXLAS:HeatI HARDE:R(1/Z HARDE:I(1/Z units K W value 3.00E+02 3.0 .00 .00 Potential TARGETS still available are: Addr Seg:Par-Type Current Value 5f HXLAST:Est-T = 200.0 K
Running this case produced the following .DAT file:
-= Hofler's 1986 thermoacoustic refrigerator =- frequency= 499.189Hz mean pressure= 1.000E+06Pa T(K) p(Pa) U(m^3/s) hdot(W) wdot(W) 300.7 30000. 0.0 0.00051 0.00000 7.6 7.65 ENDCAP Driver end 1 300.7 30000. 0.0 0.00051 0.00000 7.6 7.61 ISODUCT Room temp duct Duct wavvec =( 3.09 , -1.301E-02) m^-1 300.7 29740. -93.7 0.00049 -0.00273 7.5 7.46 HXFRST Room temp duct heat Heat exch wavvec =( 3.67 , -0.890 ) m^-1 Heat = -9.641 (W) metal temp= 300.000 Kelvin 300.7 29573. -69.2 0.00044 -0.00302 -2.2 6.65 STKSLAB Stack 4 218.6 26127. 641.5 0.00025 -0.00678 -2.2 1.04 HXLAST Cold HX 5 Heat exch wavvec =( 4.03 , -0.506 ) m^-1 Heat = 3.000 (W) metal temp= 218.862 Kelvin 218.6 25948. 662.7 0.00024 -0.00689 0.8 0.82 ISODUCT Cold Duct 6 Duct wavvec =( 3.63 , -2.005E-02) m^-1 218.6 1752. 23.3 0.00027 -0.00862 0.1 0.14 ISOCONE 7 218.6 -4226. -128.5 0.00024 -0.00843 0.0 0.03 COMPLIAN End Bulb 8 218.6 -4226. -128.5 0.00000 0.00000 0.0 0.00 HARDEND 9 inverse impedance (rho a U/p A)=( 1.173E-05, 1.913E-05) 218.6 -4226. -128.5 0.00000 0.00000 0.0 0.00 0.00000 0.0 0.00
Close examination of this result for reasonableness reveals a problem: The stack is pumping 2.2 W of enthalpy uphill, but 3.0 W of heat is being removed from the cold heat exchanger! How can this be? The problem is in our use of ISODUct and ISOCOne in the cold portion of the apparatus. DELTAE assumes that these segments are held isothermal by external means. In this case, that means that in the duct, cone, and compliance, where 0.82 W of work is dissipated into heat, some external means removes that heat. In Hofler's work, that external means was a good thermal connection between these parts and the cold heat exchanger, so that this heat appeared as a load on the cold heat exchanger.
There are two ways to deal with this problem. The first is to simply subtract the 0.82 W from the 3 W when we want to know the ``actual" net refrigeration power available at the cold heat exchanger. This is not very elegant. The second is to use the insulated segment types INSDUct and INSC0ne. (DELTAE will insulate the compliance as well.) This will force heat dissipated in these segments to show up in the nearest heat exchanger.
To do this, we use a text editor to edit the input file, changing the two segments from ISO- to INS-. We also chose to cut the cold heat exchanger heat from 3 W to 2.18 W so that the ``true" cooling power would be the same as above. Running this case produced
Thus, the work dissipated in the cold portion showed up automatically in the cold heat exchanger.-= Hofler's 1986 thermoacoustic refrigerator =- frequency= 499.163Hz mean pressure= 1.000E+06Pa T(K) p(Pa) U(m^3/s) hdot(W) wdot(W) 300.7 30000. 0.0 0.00051 0.00000 7.6 7.64 ENDCAP Driver end 1 300.7 30000. 0.0 0.00051 0.00000 7.6 7.61 ISODUCT Room temp duct Duct wavvec =( 3.09 , -1.301E-02) m^-1 300.7 29740. -93.6 0.00049 -0.00273 7.5 7.45 HXFRST Room temp duct heat Heat exch wavvec =( 3.67 , -0.890 ) m^-1 Heat = -9.632 (W) metal temp= 300.000 Kelvin 300.7 29573. -69.1 0.00044 -0.00302 -2.2 6.64 STKSLAB Stack 4 218.6 26127. 641.7 0.00025 -0.00678 -2.2 1.04 HXLAST Cold HX 5 Heat exch wavvec =( 4.03 , -0.505 ) m^-1 Heat = 2.180 (W) metal temp= 218.756 Kelvin 218.6 25948. 662.8 0.00024 -0.00689 0.0 0.81 INSDUCT Cold Duct 6 Duct wavvec =( 3.63 , -2.005E-02) m^-1 218.6 1752. 23.9 0.00027 -0.00862 0.0 0.14 INSCONE 7 218.6 -4227. -127.9 0.00024 -0.00843 0.0 0.02 COMPLIAN End Bulb 8 218.6 -4227. -127.9 0.00000 0.00000 0.0 0.00 HARDEND 9 inverse impedance (rho a U/p A)=( 9.968E-09, 2.382E-07) 218.6 -4227. -127.9 0.00000 0.00000 0.0 0.00
(INSulated segments are still under development and they don't always do what we want them to do. When using them, carefully examine the results for reasonableness. See Chapter V for details.)
To generate plots for comparison to Hofler's data, we retrun to ISODUct and ISOCOne because he added the dissipation in these components to his applied heat load for plotting. We let the heat at the cold heat exchanger be the independent variable, ranging from 2 to 8 W in 0.5 W steps. To plot the temperature ratio and the coefficient of performance (COP) relative to Carnot's COP, we include work at segment 1, T_c, and T_h in the list of plotted variables:
(The first three dependent variables listed are unclearable defaults that we ignore.) After running this case, we changed |p| to 0.015 of p_m, changed the range of Q_c to 0.7 to 3.7 W in steps of 0.5 W, and ran it again. Exiting DELTAE, we found the following .des and .plt files for the first case:Dependent Variables (outputs): PLOTS 0A 0B 0C 0D 1F 3H 5H name BEGIN:Freq. BEGIN:T-beg BEGIN:|U|@0 BEGIN:HeatI ENDCA:Work HXFRS:Metal HXLAS: units Hz K m^3/s W W K K Indpendent Variables (inputs): Outer loop: 5e HXLAS:HeatI Beg= 2.0 End= 8.0 Step= 0.50
HXLAS:HeatI BEGIN:Freq .BEGIN:T-beg BEGIN:|U|@0 ENDCA:Work HXFRS:Metal
HXLAS:Metal
W Hz K m^3/s W K K
5e 0A 0B 0C 1F 3H 5H
2.000 493.1 300.6 4.7359E-04 7.069 300.0 212.9
2.500 496.2 300.6 4.9203E-04 7.346 300.0 215.9
3.000 499.2 300.7 5.1039E-04 7.321 300.0 218.9
3.500 502.1 300.7 5.2869E-04 7.896 300.0 221.8
.
.
.
Reading this file
into spreadsheet/graphics software, and forming T_c/T_h and COPR yielded
the plots in Figs. III.4 and III.5.
These plots come reasonably close
to the measurements presented in Figs. 16 and 17 of Hofler's thesis.
Figure III.4 & III.5: Hofler refrigerator: p = 0.03 p_m. Lines are DELTAE results; points are from experimental data. Circles, p = 0.015 p_m. Squares, p = 0.03 p_m.
Returning to INS-, we now use this example to introduce some more new segment types.
Since we frequently find it useful to consider engine efficiency or refrigerator coefficient of performance (COP), normalized by their Carnot values, we have special segments COPRTarget and EFFRTarget to compute them. We can even use them as targets if desired. We added a COPRTarget to our input file:
(skipping the first segments, which we've seen before)
!------------------------------------------------------------------------
COMPLIAN End Bulb 8
4.900E-02 a Area m^2 4.229E+03 A |p| Pa
1.060E-03 b Volum m^3 -178. B Ph(p) deg
2.853E-08 C |U| m^3/s
-90.4 D Ph(U) deg
2.234E-06 E Hdot W
2.234E-06 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
HARDEND 9
0.000 a R(1/Z) = 9G? 4.229E+03 A |p| Pa
0.000 b I(1/Z) = 9H? -178. B Ph(p) deg
2.853E-08 C |U| m^3/s
-90.4 D Ph(U) deg
2.234E-06 E Hdot W
2.234E-06 F Work W
9.767E-09 G R(1/Z)
2.636E-07 H I(1/Z)
sameas 0 Gas type
ideal Solid type
COPRT COP/COP-Carnot
0.13 Target
5G NumAdr
1F DenomAdr
3H ThAdr
5H TcAdr
! The restart information below was generated by a previous run
! You may wish to delete this information before starting a run
! where you will (interactively) specify a different iteration
! mode. Edit this data only if you really know your model!
INVARS 4 0 2 0 3 0 6 3 5
TARGS 4 3 6 9 1 9 2 10 1
SPECIALS 0
Here, the COPRTarget segment directly calculates the result that we calculated
laboriously in our spreadsheet/graphics software using data from the .plt file. The
result appears in the .out file and the .dat file, and can be tabulated in
the .plt
file.
COPR can be used as a target. For instance, if we change the vector summary to
and run the code, we can find an operating point on the plots above that corresponds to COPR = 0.13.!------------------------------------------------------------------------ Iteration Vectors Summary: GUESS 0b 0c 0f 3e name BEGIN:Freq. BEGIN:T-beg BEGIN:|U|@0 HXFRS:HeatI value 4.97E+02 3.01E+02 5.06E-04 -9.6 units Hz K m^3/s W TARGET 3f 9a 9b 10a name HXFRS:Est-T HARDE:R(1/Z HARDE:I(1/Z COPRT:Targe units K value 3.00E+02 .00 .00 .13 Potential TARGETS still available are: Addr Seg:Par-Type Current Value 5e HXLAST:HeatIn= 2.190 W 5f HXLAST:Est-T = 200.0 K
Finally, a more realistic driver was added to the system, using VSPEAKer. We edited the input file, adding VSPEAKer near the beginning. We deleted the ENDCAp segment that was near the beginning because VSPEAker accounts for the oscillatory pressurization losses on its surface area. We also added a difference target DIFFTarget at the end.
TITLE Hofler's 1986 thermoacoustic refrigerator, w speaker
BEGIN 0
1.000E+06 a Mean P Pa
500. b Freq. Hz
300. c T-beg K
3.000E+04 d |p|@0 Pa
150.0 e Ph(p)0 deg
.000 f |U|@0 m^3/s
.000 g Ph(U)0 deg
helium Gas type
ideal Solid type
VSPEAKER 1
6.000E-04 a Area m^2
6.00 b R ohms
.000 c L H
8.00 d B x L T-m
5.000E-03 e M kg
.000 f K N/m
.000 g Rm N-s/m
20. h AplVol V
SAMEAS 0 Gas type
ideal Solid type
ISODUCT room temp duct 2
1.134E-03 a Area m^2
.119 b Perim m
4.260E-02 c Length m
SAMEAS 0 Gas type
ideal Solid type
HXFRST room temp heat excha 3
SAMEAS 2a a Area m^2
.600 b GasA/A
6.350E-03 c Length m
1.900E-04 d y0 m
-10. e HeatIn W
300. f Est-T K
SAMEAS 0 Gas type
ideal Solid type
STKSLAB Stack 4
SAMEAS 2a a Area m^2
.724 b GasA/A
7.850E-02 c Length m
1.800E-04 d y0 m
4.000E-05 e Lplate m
SAMEAS 0 Gas type
kapton Solid type
HXLAST Cold heat exchanger 5
SAMEAS 2a a Area m^2
.670 b GasA/A
2.540E-03 c Length m
2.550E-04 d y0 m
2.19 e HeatIn W
200. f Est-T K
SAMEAS 0 Gas type
ideal Solid type
INSDUCT Cold duct 6
3.840E-04 a Area m^2
6.940E-02 b Perim m
.167 c Length m
SAMEAS 0 Gas type
ideal Solid type
INSCONE 7
SAMEAS 6a a AreaI m^2
SAMEAS 6b b PerimI m
6.680E-02 c Length m
1.160E-03 d AreaF m^2
.121 e PerimF m
SAMEAS 0 Gas type
ideal Solid type
COMPLIANCE end bulb 8
4.900E-02 a Area m^2
1.060E-03 b Volum m^3
SAMEAS 0 Gas type
ideal Solid type
HARDEND 9
.000 a R(1/Z)
.000 b I(1/Z)
SAMEAS 0 Gas type
ideal Solid type
DIFFTARGET 10
.000 a Target
1B b
1L c
The mass, resistance, and force constant for the speaker roughly reflect the
values given in Hofler's thesis. We estimate it will take about 20 V to drive
it.
We used the difference target DIFFTarget segment to maintain resonance, by ensuring that the phases of p and U are equal at the driver. We did this by forcing their difference, computed by subtracting the values addressed by lines 10b and 10c, to be zero, the value given in line 10a. Examination of a VSPEAKer segment output
VSPEAKER 1
6.000E-04 a Area m^2 3.000E+04 A |p| Pa
6.00 b R ohms 154. B Ph(p) deg
0.000 c L H 5.074E-04 C |U| m^3/s
8.00 d B x L T-m 154. D Ph(U) deg
5.000E-03 e M kg 7.61 E Hdot W
0.000 f K N/m 7.61 F Work W
0.000 g Rm N-s/m 31.1 G WorkIn W
22.6 h AplVol V G 22.6 H Volts V
2.80 I Amps V
10.3 J Ph(Ze) deg
5.086E-04 K |Ux| m^3/s
154. L Ph(-Ux deg
sameas 0 Gas type
ideal Solid type
shows us that lines 10b and c should contain addresses 1B and 1L.
Running DELTAE with this input file, we modified guesses and targets to arrive at
Iteration Vectors Summary: GUESS 0b 0c 0e 1h 3e name BEGIN:Freq. BEGIN:T-beg BEGIN:Ph(p) VSPEA:AplVo HXFRS:HeatI value 5.00E+02 3.00E+02 150. 20. -10.0 units Hz K deg V W TARGET 3f 5e 9a 9b 10a name HXFRS:Est-T HXLAS:HeatI HARDE:R(1/Z HARDE:I(1/Z DIFFT:Targe units K W value 3.00E+02 2.2 .00 .00 .00 Potential TARGETS still available are: Addr Seg:Par-Type Current Value 5f HXLAST:Est-T = 200.0 K
This shows our five-dimensional search. It is the most complicated vector summary table we have yet encountered, so we pause to discuss how we chose our vectors. We arrived at this selection because we definitely needed the two HARDEnd impedances in the target vector (there is no hole in the end of the apparatus). Experimentally, we maintain the hot heat exchanger at 300 Kelvin; but DELTAE computes that as a result of each integration pass, so it must also be a target. Experimentally, we control the heat load on the cold heat exchanger, but because this is an HXLASt, DELTAE calculates it as a result of each integration pass, so it too must be a target. So far we have four targets, so we require guesses. Look first at the BEGIN segment for candidate guesses. Clearly the beginning temperature should be a guess: we need to guess beginning T to arrive at HXFRSt T correctly. Next, we must guess the frequency to maintain resonance. But how is resonance determined experimentally? By comparing the phases of p and U at the driver: hence, we added their difference = 0 as a fifth target. We need the phase of p at the beginning to be a guess, since the phase of everything is determined relative to that of the speaker voltage phase, which is fixed at 0 degrees. Other good candidate guesses are heats in HXFRSts or HXMIDls. The heat in the first heat exchanger must be guessed because we don't control it experimentally yet it is required by DELTAE in each pass. By now we had five targets and guesses; we needed one more guess. Our guess could have been |p| at the beginning, which would be an experimental result if we controlled the drive voltage. Instead, however, we let the drive voltage be the guess because the experimenter used it to get p to be 0.03 p_m.
Choosing the vector members is not easy for a complicated thermoacoustic system. To choose them wisely, there is no substitute for careful thought about the system and what you want it to do. We offer a few general guides for this careful thought process. It is helpful to think about what variables are (or could be, in principle) experimentally controlled and what variables are experimentally observed. These must be compared with the variables that DELTAE needs as inputs during each integration pass through the system and those that DELTAE computes as results during each integration pass.
Note that our definition of an experimental result is more general than usual. In the Hofler refrigerator case, we considered the drive voltage an experimental result because it is determined experimentally by the condition that the pressure amplitude have the desired value. We would also include in this category results of gedanken experiments, such as the length of a duct that is required to achieve resonance at a given frequency.
Hence, another useful way to think about it is represented by the following table:
Now we return to our example. Running this case produces the following .dat file:
-= Hofler's 1986 thermoacoustic refrigerator, w speaker =- frequency= 499.225Hz mean pressure= 1.000E+06Pa T(K) p(Pa) U(m^3/s) hdot(W) wdot(W) 300.7 -26922. 13236.0 0.00000 0.00000 0.0 0.00 VSPEAKER ( 22.6 , 0.000 ) Volts,( 2.75 , 0.499 ) Amps 300.7 -26922. 13236.0 -0.00046 0.00022 7.6 7.61 ISODUCT Room temp duct Duct wavvec =( 3.09 , -1.301E-02) m^-1 300.7 -26648. 13205.6 0.00076 0.00267 7.5 7.46 HXFRST Room temp duct heat Heat exch wavvec =( 3.67 , -0.890 ) m^-1 Heat = -9.648 (W) metal temp= 300.000 Kelvin 300.7 -26508. 13109.6 0.00093 0.00290 -2.2 6.65 STKSLAB Stack 4 218.6 -23730. 10951.3 0.00277 0.00620 -2.2 1.04 HXLAST Cold HX 5 Heat exch wavvec =( 4.03 , -0.506 ) m^-1 Heat = 2.190 (W) metal temp= 218.817 Kelvin 218.6 -23578. 10853.3 0.00283 0.00629 0.0 0.81 INSDUCT Cold Duct 6 Duct wavvec =( 3.63 , -2.006E-02) m^-1 218.6 -1582. 751.3 0.00356 0.00786 0.0 0.14 INSCONE 7 218.6 3850. -1750.1 0.00350 0.00768 0.0 0.02 COMPLIAN End Bulb 8 218.6 3850. -1750.1 0.00000 0.00000 0.0 0.00 HARDEND 9 inverse impedance (rho a U/p A)=( 1.011E-06, -2.571E-07) 218.6 3850. -1750.1 0.00000 0.00000 0.0 0.00 DIFFTARGET10 Derived difference = 7.137E-04 218.6 3850. -1750.1 0.00000 0.00000 0.0 0.00
This run also produces the following .out file:
TITLE Hofler's 1986 thermoacoustic refrigerator, w
speaker
!------------------------------------------------------------------------
BEGIN 0 0
1.000E+06 a Mean P Pa 499. A Freq. G( 0b) P
499. b Freq. Hz G 301. B T-beg G( 0c) P
301. c T-beg K G 154. C Ph(p)0 G( 0e) P
3.000E+04 d |p|@0 Pa 22.6 D AplVol G( 1h) P
154. e Ph(p)0 deg G -9.65 E HeatIn G( 3e) P
0.000 f |U|@0 m^3/s
0.000 g Ph(U)0 deg
helium Gas type
ideal Solid type
!------------------------------------------------------------------------
VSPEAKER 1
6.000E-04 a Area m^2 3.000E+04 A |p| Pa
6.00 b R ohms 154. B Ph(p) deg
0.000 c L H 5.074E-04 C |U| m^3/s
8.00 d B x L T-m 154. D Ph(U) deg
5.000E-03 e M kg 7.61 E Hdot W
0.000 f K N/m 7.61 F Work W
0.000 g Rm N-s/m 31.1 G WorkIn W
22.6 h AplVol V G 22.6 H Volts V
2.80 I Amps V
10.3 J Ph(Ze) deg
5.086E-04 K |Ux| m^3/s
154. L Ph(-Ux deg
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
ISODUCT Room temp duct 2
1.134E-03 a Area m^2 2.974E+04 A |p| Pa
0.119 b Perim m 154. B Ph(p) deg
4.260E-02 c Length m 2.774E-03 C |U| m^3/s
74.1 D Ph(U) deg
7.46 E Hdot W
7.46 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
HXFRST Room temp duct heat 3
sameas 2a a Area m^2 2.957E+04 A |p| Pa
0.600 b GasA/A 154. B Ph(p) deg
6.350E-03 c Length m 3.051E-03 C |U| m^3/s
1.900E-04 d y0 m 72.2 D Ph(U) deg
-9.65 e HeatIn W G -2.19 E Hdot W
300. f Est-T K = 3H? 6.65 F Work W
-9.65 G Heat W
300. H MetalT K
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
STKSLAB Stack 4 4
sameas 2a a Area m^2 2.613E+04 A |p| Pa
0.724 b GasA/A 155. B Ph(p) deg
7.850E-02 c Length m 6.790E-03 C |U| m^3/s
1.800E-04 d y0 m 65.9 D Ph(U) deg
4.000E-05 e Lplate m -2.19 E Hdot W
1.04 F Work W
301. G T-beg K
219. H T-end K
-5.61 I StkWrk W
sameas 0 Gas type
kapton Solid type
!------------------------------------------------------------------------
HXLAST Cold HX 5 5
sameas 2a a Area m^2 2.596E+04 A |p| Pa
0.670 b GasA/A 155. B Ph(p) deg
2.540E-03 c Length m 6.895E-03 C |U| m^3/s
2.550E-04 d y0 m 65.8 D Ph(U) deg
2.19 e HeatIn W = 5G? 0.000 E Hdot W
200. f Est-T K (t) 0.814 F Work W
2.19 G Heat W
219. H MetalT K
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
INSDUCT Cold Duct 6 6
3.840E-04 a Area m^2 1.752E+03 A |p| Pa
6.940E-02 b Perim m 155. B Ph(p) deg
0.167 c Length m 8.628E-03 C |U| m^3/s
65.6 D Ph(U) deg
0.000 E Hdot W
0.136 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
INSCONE 7 7
sameas 6a a AreaI m^2 4.229E+03 A |p| Pa
sameas 6b b PerimI m -24.4 B Ph(p) deg
6.680E-02 c Length m 8.436E-03 C |U| m^3/s
1.160E-03 d AreaF m^2 65.5 D Ph(U) deg
0.121 e PerimF m 0.000 E Hdot W
2.286E-02 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
COMPLIAN End Bulb 8 8
4.900E-02 a Area m^2 4.229E+03 A |p| Pa
1.060E-03 b Volum m^3 -24.4 B Ph(p) deg
1.128E-07 C |U| m^3/s
-38.7 D Ph(U) deg
2.311E-04 E Hdot W
2.311E-04 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
HARDEND 9 9
0.000 a R(1/Z) = 9G? 4.229E+03 A |p| Pa
0.000 b I(1/Z) = 9H? -24.4 B Ph(p) deg
1.128E-07 C |U| m^3/s
-38.7 D Ph(U) deg
2.311E-04 E Hdot W
2.311E-04 F Work W
1.011E-06 G R(1/Z)
-2.571E-07 H I(1/Z)
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
DIFFTARGET 10 10
0.000 a Ph(a-b =10A? 7.137E-04 A Ph(a-b
1B b D1Addr
1L c D2Addr
! The restart information below was generated by a previous run
! You may wish to delete this information before starting a run
! where you will (interactively) specify a different iteration
! mode. Edit this data only if you really know your model!
INVARS 5 0 2 0 3 0 5 1 8 3 5
TARGS 5 3 6 5 5 9 1 9 2 10 1
SPECIALS 0
These acoustic and thermal results are the same as for without the speaker, except that everything is shifted in phase by -26 degrees. This shift occurred because we had set the phase, arbitrarily, at zero before, but this time the speaker voltage determined the zero phase for the system, and the nonzero imaginary part of its mechanical impedance caused a phase shift between the voltage and the velocity. New results appear in the VSPEAker segment; note for example that [tex2html_wrap2602]is the difference between the work into the segment ( [tex2html_wrap2604]) and the work out of it.
