Suppose we take a shower and collect 100 F greywater in the upper part of
a $30 100'x4" black plastic corrugated drainpipe coil containing 3 $20
100'x1" pieces of black plastic polyethylene pipe, with bidirectional
plug flow, like this, viewed in a fixed font like Courier:

shower
in
| ----------------------------------------- hot water to shower
| | Tl |
--------- sewer ---------
| | Tg | out | 120F |
| | | | |
| | | ^ | |
| | | | | |
| |1" |4" | | tank |
| | | | | water |
| | | | | heater |
| | | | | |
| | | | | |
| | | | | |
| | |---- | |
| | | | 55F |
--------- P ---------
| ---- Tc |
-----------------| - |------------------- cold water supply
----

Now we disable the water heater and run a slow, low-power pump P
(eg Grainger's $120 4PC86 (Taco 003-BC4-2) 1/40 HP 120V 0.43A pump)
if Tg - Tl 5 F and Tg - Tc 5 F, and enable the water heater again
when Tg - Tc 5 F...

Is this a GFX-Star? It's hard to tell from the website description.

20 UPIPE=78.5'U-value of 10' section of 3 1" pipes (Btu/h-F)
30 CFRESH=1.25*8.33'thermal capacitance of 10' of fresh water (Btu/F)
40 VGREY=10*3.14159*(2/12)^2'volume of 10' of greywater (ft^3)
50 CGREY=VGREY*62.33-CFRESH'thermal capacitance of 10' of greywater (Btu/F)
60 CSERIES=CFRESH*CGREY/(CFRESH+CGREY)'caps in series (Btu/F)
70 RC=CSERIES/UPIPE'combined time constant (hours)
80 EXPF=EXP(-1/60/RC)'exponential factor
90 FOR SHOWER = 1 TO 100'simulate showers
100 FOR M=0 TO 89'simulate 10 min shower every 90 minutes
110 IF M9 GOTO 220'rest vs shower
120 IF SHOWER 100 GOTO 150
140 PRINT 400+M;"'";M,TG(9)
150 TGT=TG(0)'save original Tg(0) for later Tg(1) calc
160 TG(0)=(100*CFRESH+TG(0)*(CGREY-CFRESH))/CGREY'move greywater in
170 FOR S=1 TO 9'pipe section (9-fresh water in and greywater out)
180 TGP=TG(S)'buffer
190 TG(S)=(TGT*CFRESH+TG(S)*(CGREY-CFRESH))/CGREY'move greywater down
200 TGT=TGP'buffer
210 NEXT S
220 IF (TG(0)-TF(0))5 OR TG(0)60 GOTO 290'no pumping
230 PUMP=PUMP+1'pump in fresh water at bottom
240 IF SHOWER49 THEN HEAT=HEAT+CFRESH*(TF(0)-55)'gain from gw
250 FOR S=0 TO 8'shift fresh water up
260 TF(S)=TF(S+1)
270 NEXT S
280 TF(9)=55'move cold water in at the bottom
290 FOR S=0 TO 9'rest
300 TFINAL=(TF(S)*CFRESH+TG(S)*CGREY)/(CFRESH+CGREY)
310 TF(S)=TFINAL+(TF(S)-TFINAL)*EXPF'new fresh temp (F)
320 TG(S)=TFINAL+(TG(S)-TFINAL)*EXPF'new grey temp (F)
330 NEXT S
340 NEXT M
350 NEXT SHOWER
360 SHOWERGY=50*10*CFRESH*(100-55)'50 showers with no GWHX (Btu)
370 PRINT HEAT,SHOWERGY,HEAT/SHOWERGY,PUMP

time gwout
(min) (F)

0 55.50439
1 55.66506
2 55.83802
3 56.02375
4 56.22276
5 56.43565
6 56.66306
7 56.90572
8 57.16441
9 57.43995

231607 234281.3 .9885854 1458

Looks good on paper, with 98.9% heat recovery :-)

Nick