Yes. Thanks for noticing.
Most of the cold water flowing into the heat exchanger comes from the tank
bottom, since most pumping (about 30 Cfresh bursts over 2 hours) happens
during non-shower times, in order to keep the heat exchanger usefully
employed with significant temp diffs more of the time.
Run the pump to recover heat from warm greywater that just recently entered
the 4" pipe and stopped flowing? i.e., run the pump right *after* a shower
to recover heat from the greywater that is now stagnant in the drain pipe?
But not *during* the shower?
So during a shower, does cold water flow into the tank alone and not through
the heat-exchanger? Or is the 'hot-water to shower' a mixture of two
parallel paths, one through the tank and the other through the idle pump and
the heat-exchanger? From your code, it looks like there is no freshwater
flow throught the hx during the shower, but I don't see what prevents this
since TF(0) will heat up to TG(0) when the pump is stopped, and rapidly cool
when the pump is started.
There might be no
fresh water flow into the heat exchanger during a shower. This is not
the same as running a pump to increase fluid velocity and film
conductance.
I imagine the tank bottom temp will be close to 55 F, with a slow pump and
water heater stratification, with the lower heating element disabled.
Here's the result with the pipe volume fix, and another in line 100,
which didn't have the NP multiplier.
20 DIM TF(50),TG(50)
I=4*ATN(1)
30 VG=1.25*8.33/62.33'volume of 1.25 gallons (ft^3)
40 V4=PI*(2/12)^2'volume of 1' of 4" pipe (ft^3)
50 NP=3'number of 1" pipes
60 V1=NP*PI*(.5/12)^2'volume of 1' of 1" pipes (ft^3)
70 LS=VG/(V4-V1)'simulation segment length (feet)
80 NS=INT(100/LS+.5)'number of simulation segments
100 UPIPE=10*LS*NP*PI/12'U-value of L' section of 1" pipe (Btu/h-F)
110 CFRESH=LS*V1*62.33'thermal capacitance of L' of 1" pipe (Btu/F)
120 CGREY=VG*62.33'capacitance of L' of greywater (Btu/F)
130 CSERIES=CFRESH*CGREY/(CFRESH+CGREY)'caps in series (Btu/F)
140 RC=CSERIES/UPIPE'combined time constant (hours)
150 EXPF=EXP(-1/60/RC)'exponential factor
160 FOR SHOWER = 1 TO 500'simulate showers
170 FOR M=0 TO 119'simulate 10 min shower every 120 minutes
180 IF M>9 GOTO 250'rest vs shower
190 IF SHOWER <500 GOTO 210
200 PRINT 600+M;"'";M,TG(NS-1)
210 FOR S=NS-1 TO 1 STEP -1'pipe section (ns-1<->fw in and gw out)
220 TG(S)=TG(S-1)'move greywater down
230 NEXT S
240 TG(0)=100'move greywater in
250 IF (TG(0)-TF(0))>5 OR TG(0)<60 GOTO 310'no pumping
260 IF SHOWER>249 THEN HEAT=HEAT+CFRESH*(TF(0)-55)
UMP=PUMP+1
270 FOR S=0 TO NS-2'shift fresh water up
280 TF(S)=TF(S+1)
290 NEXT S
300 TF(NS-1)=55'move cold water in at the bottom
310 FOR S=0 TO NS-1'rest
320 TFINAL=(TF(S)*CFRESH+TG(S)*CGREY)/(CFRESH+CGREY)
330 TF(S)=TFINAL+(TF(S)-TFINAL)*EXPF'new fresh temp (F)
340 TG(S)=TFINAL+(TG(S)-TFINAL)*EXPF'new grey temp (F)
350 NEXT S
360 NEXT M
370 NEXT SHOWER
380 SHOWERGY=250*10*CGREY*(100-55)'250 showers with no GWHX (Btu)
390 PRINT 500+NP;"'";HEAT,SHOWERGY,HEAT/SHOWERGY,PUMP
number of recovered required recovered pumping
pipes heat (Btu) heat (Btu) fraction events
1 175586.9 234281.3 .7494707 6095
2 229994.3 234281.3 .9817018 4365
3 1168364 1171406 .997403 15060
That third column doesn't make sense. The heat required for a ten minute
shower of 100F water from 55F water should be the same in all three cases.
VG and CGREY are constant regardless of the number of pipes, so SHOWERGY
should also be a constant regardless of the number of pipes. Something very
'fishy' about the cases with one or two pipes are identical, yet magically
the third case changes to a smaller value. I calculate the total energy
needed for such a shower is 1171406 BTU (the same as the 3-pipe case). Yet
your calculations for the 1 and 2 pipe case claim to have recovered more
energy than the total energy required (implying an efficiency greater than
100%). Looks like more GIGO.
The very large number of pumping events for the 3 pipe case suggests that
the 'pump' is cycling every other iteration because the TF(0) temperature is
jumping up and down. That averages to 60 times for each 'shower interval',
every two minutes. This suggests to me that the model is unstable and needs
finer time-steps. But I don't see an explicit value for pump flow, so it
seems the model is assuming the same 1.25 gpm. That is much too high as it
is cooling TF(0) so much that it cycles the pump off each time the pump
starts. I thought your idea was to use a very low flow pump to avoid mixing
in the tank?
The only way that TF(NS-1) would always be 55F is if the tank were to stay
well stratified. And no flow through the hx when the shower is on, only
when the pump is on *after* the shower turns off. Tank stratification and a
'perforated dip-tube' would be necessary so the warm freshwater out of the
hx when the pump is running enters into the tank at the correct elevation.
Entering directly in the top would trigger the upper heating element, and
entering through a conventional, solid dip-tube would warm the bottom of the
tank excessively, reducing the heat that can be recovered in the hx by
raising freshwater inlet temperature.
Putting NP in line 100 raised the 2-pipe fraction a lot, but that would
drop
with shorter shower intervals. I increased the number of 3-pipe showers
from
100 to 500 because its fraction came out greater than 1, given the
stochastic
pumping. The greywater outlet seems surprisingly warm during the last
shower.
The 3-pipe simulation only has 13 pipe sections. It may need more.
min gw outlet
0 59.37316 F
1 60.09058
2 60.88006
3 61.63319
4 62.3915
5 63.15523
6 63.91952
7 64.68189
8 65.44382
9 66.20461
Yeah, there's definitely something wrong with all this. With those gw
outlet temperatures, you have....
10.4125*(59.37316-55) = 45.5 Btu
10.4125*(60.09058-55) = 53.0 Btu
10.4125*(60.88006-55) = 61.2 Btu
10.4125*(61.63319-55) = 69.1 Btu
10.4125*(62.39150-55) = 77.0 Btu
10.4125*(63.15523-55) = 84.9 Btu
10.4125*(63.91952-55) = 92.9 Btu
10.4125*(64.68189-55) = 100.8 Btu
10.4125*(65.44382-55) = 108.7 Btu
10.4125*(66.20461-55) = 116.7 Btu
for a total of 809.8 Btu going down the drain as greywater, yet your
calculations show a loss of (1171406-1168364)= 3042 Btu. Something
somewhere doesn't add up. You may need to go with shorter time steps than
one minute, and find a different way to control the cycling of the pump or
reduce its flow rate considerably.
daestrom
