The 5-inch engine is described in detail in J. Acoust. Soc. Am. 92, 1551 (1992). The device described there is used to illustrate the fully thermoacoustic capabilities of DELTAE here and to reproduce some of the figures in that paper.
The apparatus looks like the following.
Figure III.1: 5-inch engine.
Beginning with the input file (5inch.in, in the examples directory) to illustrate stack and heat exchanger segment types:
TITLE Five-Inch Thermoacoustic Engine BEGIN Initial 13.8e5 (Pa) mean pressure 100. (Hz) freq 500. Starting Temp 8.e4 Mag(Pa) acoustic pressure @x=0 0. Phase (deg) acoustic pressure @x=0 0. Mag(vdot) vol. veloc @x=0 0. Phase (deg) vol. veloc @x=0 helium gas type ENDCAP Hot End 0.01292 (m^2) area sameas 0 gas type ISODUCT Hot Duct sameas 1 (m^2) total area 0.403 (m) perim 0.279 (m) length sameas 0 gas type HXFRST Hot HX sameas 1 (m^2) total area 0.393 gas area/total area 0.060 (m) length 0.483e-3 (m) y0 2210.20 (W) heat 550. (K) temperature sameas 0 gas type STKCIRC Honeycomb Stack sameas 1 (m^2) total area 0.81 gas area/total area 0.279 (m) length 0.50e-3 (m) r0 0.05e-3 (m) L:half of sht thcknss sameas 0 gas type stainless stack material HXLAST Cold HX 0.01267 (m^2) total area 0.486 gas area/total area 0.0508 (m) length 0.406e-3 (m) y0 0.0 (W) heat 303. (K) temperature sameas 0 gas type ISODUCT Cold Duct sameas 5 (m^2) total area 0.399 (m) perim 3.654 (m) length sameas 0 gas type ENDCAP Cold End sameas 5 (m^2) area sameas 0 gas type HARDEND 0. Re(zinv) 0. Re(zinv) sameas 0 gas type
Of the three types of heat exchanger segments, only two are shown here: HXFRSt comes before a stack; HXLASt comes after a stack; HXMIDl comes between stacks (as in the old beercooler). They differ in whether the heat flow is considered to be an input (possibly a guess) or a result.
All HX's are assumed to have parallel-plate geometry, with plate spacing 2y0. Other geometry is given in straightforward format. Wave propagation through heat exchangers is computed using a complex wavevector including both viscous and thermal dissipation in this geometry.
One additional feature of HX's is heat flow. Positive heat flows into the apparatus. In HXFRSt and HXMIDl, the heat flow determines the change in enthalpy in the heat exchanger. Thus, in these cases heat flow can be either be fixed, or it can be a member of the guess vector or an independent plot variable. In HXLASt, the change in enthalpy determines the heat flow, so the heat can be a result or target (and optionally a dependent plot variable). This example uses the hot heat exchanger's heat flow as the independent plot variable and the cold heat exchanger's heat flow as a simple result that is largely ignored here.
A second additional feature of the HX's is the temperature difference between the mean-gas temperature and the heat exchanger metal temperature, proportional to the heat flow. Its dependence on the geometry of the heat exchanger is given in Chapter V. [This temperature difference can be computed only within an accuracy of about a factor of 2, even in the acoustic approximation; nevertheless, it is include, to prevent naive users from being led to designs with heat exchangers of negligible surface area that have negligible losses and that would appear to have no disadvantages if the temperature difference were not included. Future revisions of DELTAE, hopefully, will have an accurate calculation algorithm for this effect. Meanwhile, however, if you prefer not to use this feature, use the gas mean temperature instead of the metal temperature by using a FREETarget (see Section V.A) to access the temperature in the adjacent stack segment (parameter G or H).] Metal temperature can be a target or a result. In this example,the cold heat exchanger's temperature is used as a target and the hot heat exchanger's temperature is used as a result and plotted.
Four types of stacks are allowed, STKSLab for parallel-plate geometry, STKCIrc for circular pores, STKREct for rectangular and square pores, and STKPIns for pin-array stacks. The geometry is given as shown in the example. Evolution of Tm, p, and Tm(u) are computed as described in J. Acoust. Soc. Am. 92,1551 (1992).
You can execute DELTAE using the input file above and use (C)lear|set to ask for default targets:
We selected `p' because this device is a prime mover. Examining the vector summary, we find: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)? p
Iteration Vectors Summary: GUESS 0b 0c 0d name BEGIN:Freq. BEGIN:T-beg BEGIN:|p|@0 value 1.00E+02 5.00E+02 8.00E+04 units Hz K Pa TARGET 5f 8a 8b name HXLAS:Est-T HARDE:R(1/Z HARDE:I(1/Z units K value 3.03E+02 .00 .00 Potential TARGETS still available are: Addr Seg:Par-Type Current Value 3f HXFRST:Est-T = 550.0 K 5e HXLAST:HeatIn= .0000 W
DELTAE has made good default choices for guess and target vector elements. The default is a three-dimensional search, with end impedance and cold heat-exchanger temperature as targets.
Other choices could be made for this table. For instance, the cold-duct length could be substituted for the frequency in the guess vector. A fourth component, such as the hot heat-exchanger temperature 3f could be added to the target vector and, simultaneously, the hot heat-exchanger heat3e could be added to the guess vector. For now, however, these vectors will be left as they are, since they reflect the point of view adopted in Swift,J. Acoust. Soc. Am. 1992.
If you run this case, you will get the following .dat file:
This run will also produce the following .out file:-= Five-Inch Thermoacoustic Engine =- frequency= 121.020Hz mean pressure= 1.380E+06Pa T(K) p(Pa) U(m^3/s) hdot(W) wdot(W) 557.6 73419. 0.0 0.00000 0.00000 0.0 0.00 ENDCAP Hot End 557.6 73419. 0.0 -0.00003 0.00000 -1.2 -1.22 ISODUCT Hot Duct Duct wavvec =( 0.549 , -2.010E-03) m^-1 557.6 72559. 6.9 -0.00032 -0.08744 -11.7 -11.75 HXFRST Hot HX Heat exch wavvec =( 0.669 , -0.194 ) m^-1 Heat = 2210.200 (W) metal temp= 563.256 Kelvin 557.6 71394. 482.1 -0.00202 -0.09646 2198.5 -95.28 STKCIRC Honey Stack 306.4 65521. 3146.1 0.01282 -0.15896 2198.5 169.81 HXLAST Cold HX Heat exch wavvec =( 0.858 , -0.162 ) m^-1 Heat = -2113.958 (W) metal temp= 302.996 Kelvin 306.4 62886. 3566.9 0.01214 -0.16668 84.5 84.49 ISODUCT Cold Duct Duct wavvec =( 0.740 , -1.647E-03) m^-1 306.4 -69413. -4134.8 -0.00002 -0.00002 0.7 0.66 ENDCAP Cold End 306.4 -69413. -4134.8 0.00000 -0.00002 0.0 0.02 HARDEND inverse impedance (rho a U/p A)=( 1.408E-06, 5.303E-05) 306.4 -69413. -4134.8 0.00000 -0.00002 0.0 0.02 EFFRT Carnot Efficiency Efficiency normalized by Carnot = 8.382E-02 306.4 -69413. -4134.8 0.00000 -0.00002 0.0 0.02
TITLE Five-Inch Thermoacoustic Engine
!------------------------------------------------------------------------
BEGIN Initial 0
1.380E+06 a Mean P Pa 121. A Freq. G( 0b) P
121. b Freq. Hz G 558. B T-beg G( 0c) P
558. c T-beg K G 7.342E+04 C |p|@0 G( 0d) P
7.342E+04 d |p|@0 Pa G
0.000 e Ph(p)0 deg
0.000 f |U|@0 m^3/s
0.000 g Ph(U)0 deg
helium Gas type
ideal Solid type
!------------------------------------------------------------------------
ENDCAP Hot End 1
1.292E-02 a Area m^2 7.342E+04 A |p| Pa
0.000 B Ph(p) deg
3.322E-05 C |U| m^3/s
180. D Ph(U) deg
-1.22 E Hdot W
-1.22 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
ISODUCT Hot Duct 2
sameas 1a a Area m^2 7.256E+04 A |p| Pa
0.403 b Perim m 5.476E-03 B Ph(p) deg
0.279 c Length m 8.744E-02 C |U| m^3/s
-90.2 D Ph(U) deg
-11.7 E Hdot W
-11.7 F Work W
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
HXFRST Hot HX 3
sameas 1a a Area m^2 7.140E+04 A |p| Pa
0.393 b GasA/A 0.387 B Ph(p) deg
6.000E-02 c Length m 9.648E-02 C |U| m^3/s
4.830E-04 d y0 m -91.2 D Ph(U) deg
2.210E+03 e HeatIn W 2.198E+03 E Hdot W
550. f Est-T K (t) -95.3 F Work W
2.210E+03 G Heat W
563. H MetalT K
sameas 0 Gas type
ideal Solid type
!------------------------------------------------------------------------
STKCIRC Honey Stack 4
sameas 1a a Area m^2 6.560E+04 A |p| Pa
0.810 b GasA/A 2.75 B Ph(p) deg
0.279 c Length m 0.159 C |U| m^3/s
5.000E-04 d r0 m -85.4 D Ph(U) deg
5.000E-05 e Lplate m 2.198E+03 E Hdot W
170. F Work W
558. G T-beg K
306. H T-end K
265. I StkWrk W
helium Gas type
stainless Solid type
!------------------------------------------------------------------------
HXLAST Cold HX 5
1.267E-02 a Area m^2 6.299E+04 A |p| Pa
0.486 b GasA/A 3.25 B Ph(p) deg
5.080E-02 c Length m 0.167 C |U| m^3/s
4.060E-04 d y0 m -85.8 D Ph(U) deg
0.000 e HeatIn W (t) 84.5 E Hdot W
303. f Est-T K = 5H? 84.5 F Work W
-2.114E+03 G Heat W
303. H MetalT K
helium Gas type
ideal Solid type
!------------------------------------------------------------------------
ISODUCT Cold Duct 6
sameas 5a a Area m^2 6.954E+04 A |p| Pa
0.399 b Perim m -177. B Ph(p) deg
3.65 c Length m 2.827E-05 C |U| m^3/s
-129. D Ph(U) deg
0.661 E Hdot W
0.661 F Work W
helium Gas type
ideal Solid type
!------------------------------------------------------------------------
ENDCAP Cold End 7
sameas 5a a Area m^2 6.954E+04 A |p| Pa
-177. B Ph(p) deg
2.092E-05 C |U| m^3/s
-88.1 D Ph(U) deg
1.932E-02 E Hdot W
1.932E-02 F Work W
helium Gas type
ideal Solid type
!------------------------------------------------------------------------
HARDEND 8
0.000 a R(1/Z) = 8G? 6.954E+04 A |p| Pa
0.000 b I(1/Z) = 8H? -177. B Ph(p) deg
2.092E-05 C |U| m^3/s
-88.1 D Ph(U) deg
1.932E-02 E Hdot W
1.932E-02 F Work W
1.408E-06 G R(1/Z)
5.303E-05 H I(1/Z)
helium Gas type
ideal Solid type
! 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 3 0 2 0 3 0 4
TARGS 3 5 6 8 1 8 2
SPECIALS 0
The .dat file is a segment-by-segment listing of results of the run. The three members of the guess vector (f, T_begin, and |p|_begin), which we had guessed would be near 100 Hz, 500 Kelvin, and 80,000 Pa, have turned out to be 121.020 Hz, 557.6 Kelvin, and 73,419 Pa; these values appear in the first few lines of 5inch.dat. Temperature; real and imaginary pressure and volume velocity; enthalpy flow; and work flow are listed at each transition between segments. Be sure the complex volume velocity at HARDEnd is zero, as required by two members of the target vector.
Some segments have additional information listed. Ducts and heat exchangers list wavevector (mostly real in the wide-open ducts; with large imaginary components in the much more lossy heat exchangers). Heat exchangers also list heat flow and metal temperature. Note that the metal is hotter than the gas in the hot heat exchanger, where the (positive) heat flows from metal to gas, and that the metal is cooler than the gas in the cold heat exchanger, where the (negative) heat flow is from gas to metal. Note also that DELTAE successfully hit the target metal temperature of 303 Kelvin in the cold heat exchanger.
Now examine the enthalpy and work flow columns in 5inch.dat. The hot endcap absorbs 1.2 W of work, and the hot duct absorbs 11.7 - 1.2 = 10.5 W of work. The minus signs on enthalpy and work indicate energy flows `up' the apparatus, toward the BEGINning.
The hot heat exchanger absorbs 95.28 - 11.75 = 85.53 W of work. Because 2210.2 W of heat are added through it, the enthalpy flow must increase by that amount; hence, the enthalpy flow changes from -11.7 W to 2198.5 W in the hot heat exchanger.
The enthalpy flow remains constant at 2198.5 W through the stack, which produces 169.81 - (-95.28) = 265.09 W of work. Part of that work (95.28 W) flows up to supply work to the hot parts of the engine; the rest (169.81 W) flows down to supply work to the cold parts of the engine.
An examination of the cold heat exchanger listing parallels that of the hot heat exchanger, and the cold duct and endcap parallel the hot ones.
Some of this information is also available in the .out file, but it appears in a format that can be used as an input file for subsequent runs. The .out file is also a segment-by-segment listing, with a restart table appended. In the segment-by segment listing, the variables on the left are used in the input file. They include anything that can be used as a guess or target. Anything that was used as guess or independent plot variable contains its most recent value instead of the initial value supplied be the .in file. The variables on the right can be used as dependent variables in plots and can be compared to targets. We will encounter examples of each as we examine typical segments of this file.
The left portion of the BEGIN segment is in the .in-file format. Freq, T-beg, and |p|@0 are marked with ``G" signifying their membership in the guess vector. They also appear in the right column, marked with ``P", signifying their status as default dependent plot variables. The right column of the BEGIN segment is a special case: it contains a copy of each guess vector variable with the values that were used in DELTAE's last iteration. To identify their origin, the units for each of these `output' variables are replaced by the address (e.g., ``0b") that they were copied from. This occurs only in the BEGIN segment.
Now examine the cold heat-exchanger segment. Again, the left column is the familiar input-file format. HeatIn is marked ``(t)" to indicate that it is a potential target variable, though we did not use it as such. Est-T is marked ``=5H?'' to show that it is indeed a target variable, to be compared to the computed MetalT variable that appears in the right column. For all input (left side) parameters that DELTAE recognizes as potential targets, it knows the location of the appropriate result to compare with the target value. The set of freetarget segments, each of which introduces a new target variable, allows the result to which it is compared to be flexibly defined. See Section IV.A for an introduction to freetargets.
HARDEnd has two more examples of the markers that indicate target variables. There, the target values are 0.0, and DELTAE's solution has reached 1.408e-6 and 5.303e-5, which it judges to be close enough to zero.
The restart table at the end is translated thus:
INVARS 3 0 2 0 3 0 4 means 3 variables: 0b, 0c, 0d
TARGS 3 5 6 8 1 8 2 means 3 variables: 5f, 8a, 8b
This is an encoded version of the same information that is indicated by the guess and target flags, explained above, and is visible in the vector status summary table. Here, DELTAE would find this information automatically when using this .out file as a new input file.
To plot some results for this 5-inch engine case, execute DELTAE with this file again and modify the Plot summary to be
Accomplishing this process required that we ``plot another parameter" three times to add 3H and 8A d to the dependent variable list and establish 3e as independent variable and set its initial, final, and step values. (T-beg and |p|@0 are of minor interest now, but could not be deleted from the list of plot variables because members of the guess vector appear here by default.)Dependent Variables (outputs): PLOTS 0A 0B 0C 3H 8A name BEGIN:Freq. BEGIN:T-beg BEGIN:|p|@0 HXFRS:Metal HARDE:|p| units Hz K Pa K Pa Indpendent Variables (inputs): Outer loop: 3e HXFRS:HeatI Beg= 9.50E+02 End= 50. Step= -33.
Next, we modified mean pressure to be 19.2 bar, and ran the code. When completed, we modified mean pressure to 13.8 bar, and ran it again, appending the new results to the .PLT file. Three more runs with mean pressures of 9.6, 6.9, and 5.2 bar completed the data set. We exited from DELTAE, and checked to see that it has created the .des and .plt files:
HXFRS:HeatI BEGIN:Freq. BEGIN:T-beg BEGIN:|p|@0 HXFRS:Metal HARDE:|p|
W Hz K Pa K Pa
3e 0A 0B 0C 3H 8A
950.0 120.6 562.7 5.9741E+04 566.2 5.6827E+04
916.7 120.5 562.6 5.8637E+04 566.1 5.5777E+04
883.4 120.5 562.6 5.7511E+04 566.0 5.4706E+04
850.1 120.5 562.5 5.6362E+04 565.9 5.3614E+04
816.8 120.5 562.5 5.5190E+04 565.7 5.2499E+04
.
.
.
We read this .plt file into a spreadsheet/graphics program for minimal
massaging: convert pressure amplitude at the cold end from Pascals to
bar, and then square that number; subtract 303 Kelvin from T_h, and add the
old heat leak to the room to Q_h. Plotting these results then yields the
curves shown in Fig. iii.2, resembling Figs. 5, 6, and 7 in
J. Acoust. Soc. Am. 92,1551 (1992).
These curves differ slightly from
those in the article, because of the inclusion of the small
gas-to-metal temperature differences in the heat exchangers.
Figure III.2: 5-inch engine results. Lines are DELTAE results; points are from experimental data
