jonathan_traupman Posted February 23, 2005 Share Posted February 23, 2005 <h4>Introduction</h4> <p>Last week, there was yet another thread asking which films are bestfor long exposures. As usual, I recommended my favorites, Fuji Acrosfor B&W and Velvia 100F, Provia, or Astia for color. In the ensuingdiscussion, I was asked why I didn't use color negative film,particularly since the inevitable color casts that arise when shootingat night are easier to correct in printing than by using filters withslides. <p>Generally, I've avoided color negative film for night work becausethe published corrections for reciprocity failure are much more severethan for slide film. Worse than that, the documentation for mostnegative films have very little information on reciprocity failure.(e.g. Fuji Reala is listed as +1/3 stop @ 4s, +1 stop @ 16s, and N.R.at 64s, Kodak UC100 only says no correction to 10s, after that, doyour own tests.) <p>So, confronted with a boring, rainy day off on Monday, I proceededto test both Reala and UC100 for reciprocity failure. <h4>Methodology</h4> <p>Before I start, let me state that my testing apparatus leaves quitea bit to be desired. Ideally, I'd expose the film to a light source ofknown intensity and color for a precisely timed period of time.Lacking the required equipment to do this, I improvised. <p>My test target was a gray piece of mat board propped up in myoffice. I illuminated it with the daylight coming through a southeastfacing window. The amount of illumination was controlled by adjustingthe louvered blinds on the window. Weather conditions that day wereovercast, so it probably wasn't exactly 5500K. <p>Metering was done by taking a reflected spot reading off the centerof the gray card with my Sekonic L-558 light meter. ISO was set to 100for both films. I adjusted the light until the meter indicated exactly1/4s at f/2.8 for the base exposure. I readjusted the light beforeeach exposure to ensure the light stayed constant. The largest driftin illumination from start to end of a single exposure was only around0.1 stops, which won't qualitatively affect the results very much. <p>Exposure was through a Canon EOS-3 with 24-70/2.8 lens set at 50mmand focused at infinity. The camera was tripod mounted and fired viacable release. In order to get long enough exposures, I used a Tiffen3 stop ND filter over the lens in addition to stopping it down. Theremay be a slight tint to the filter that affects color shift results,but I've never noticed anything too severe while using it in thefield. The camera was set to "M" mode for shots less than 30s, and"bulb" for shots longer than 30s. For the longer shots, I timed theexposure using a stopwatch, and think I got at least within a secondof the proper time. <p>The first frame exposured was at f/2.8 and 1/4s with no filter toestablish baseline densities. I then attached the ND filter and shotseveral frames at 4,8,15,30,60, and 120 seconds with the aperture setappropriately. At each time, I shot a baseline exposure at the meteredtime and aperture and a series of shots opening up the aperture 1/3stop at a time. <p>I measured the resulting strip of film on a Noritsu DM-201densitometer. I set the d-zero to the value of frame 1 (the referenceframe) and measured the divergence from this reference density foreach frame. <h4>Results</h4> <p>For each exposure time, I list the number of stops of additionalexposure necessary to get the visual density channel closest to thereference density. Please note that this is the necessary additionalexposure obtained by opening up the aperture. Because of the wayreciprocity failure works, you'll need more than the listed amount ofadditional exposure if you compensate by increasing the time. Thecalculation to figure out the amount of extra time needed is a bitcomplicated. <p>In some cases I give a range, indicating that one exposure was alittle thin, but the next +1/3 stop was a bit dense. I also try toguess at the color shift by looking at the RGB densities reported. <br><br><center><table><tr><td align=center><b>Metered time</b><td align=center><b>Exposurecompensation</b><td align=center><b>Color shift</b><tr><td align=center><b>4s</b><td align=center>+0-1/3<tdalign=center>yellow-green<tr><td align=center><b>8s</b><td align=center>+0-1/3<tdalign=center>yellow-green<tr><td align=center><b>15s</b><td align=center>+1/3<tdalign=center>yellow-green<tr><td align=center><b>30s</b><td align=center>+1/3<tdalign=center>yellow-red<tr><td align=center><b>60s</b><td align=center>+2/3<tdalign=center>yellow<tr><td align=center><b>120s</b><td align=center>+1<td align =center>green-blue<caption><b>Reciprocity data for Fuji Reala 100</b> </table></center> <br><br><center><table><tr><td align=center><b>Metered time</b><td align=center><b>Exposurecompensation</b><td align=center><b>Color shift</b><tr><td align=center><b>4s</b><td align=center>+1/3<tdalign=center>green-cyan<tr><td align=center><b>8s</b><td align=center>+1/3<tdalign=center>cyan-green<tr><td align=center><b>15s</b><td align=center>+2/3<tdalign=center>cyan-blue<tr><td align=center><b>30s</b><td align=center>+2/3<tdalign=center>cyan-blue<tr><td align=center><b>60s</b><td align=center>+1<tdalign=center>blue-cyan<tr><td align=center><b>120s</b><td align=center>+1-1 1/3<td align =center>blue-cyan<caption><b>Reciprocity data for Kodak UC100</b> </table></center> <br><p>since I may be misinterpreting the densitometer data (particularlyregarding the color shift) here is the relevant raw data. In somecases, I list two entries for a particular exposure time. In thesecases, there are two exposures that are approximately equally close tothe reference density. In practice, it probably makes the most senseto err on the side of overexposure. <br><br> <center><table><caption><b>Raw data for Fuji Reala 100</b></caption><tr><td align=center><b>Exposure</b><td align=center><b>V</b><tdalign=center><b>B</b><td align=center><b>G</b><td align=center><b>R</b><tr><td align=center><b>baseline 1/4s</b><td align=center>0.984<tdalign=center>1.663<td align=center>1.341<td align=center>0.928<tr><td align=center><b>4s base</b><td align=center>-0.022<tdalign=center>-0.088<td align=center>-0.024<td align=center>-0.043<tr><td align=center><b>4s + 1/3</b><td align=center>0.036<tdalign=center>-0.014<td align=center>0.052<td align=center>0.033<tr><td align=center><b>8s base</b><td align=center>-0.020<tdalign=center>-0.078<td align=center>-0.019<td align=center>-0.042<tr><td align=center><b>8s + 1/3</b><td align=center>0.034<tdalign=center>-0.005<td align=center>0.057<td align=center>0.029<tr><td align=center><b>15s +1/3</b><td align=center>0.026<tdalign=center>-0.011<td align=center>0.047<td align=center>0.015<tr><td align=center><b>30s + 1/3</b><td align=center>0.000<tdalign=center>-0.129<td align=center>-0.017<td align=center>0.000<tr><td align=center><b>60s +2/3</b><td align=center>0.008<tdalign=center>-0.078<td align=center>0.009<td align=center>0.002<tr><td align=center><b>120s +1</b><td align=center>0.037<tdalign=center>0.046<td align=center>0.075<td align=center>0.018</table></center> <br><br> <center><table><caption><b>Raw data for Kodak UC100</b></caption><tr><td align=center><b>Exposure</b><td align=center><b>V</b><tdalign=center><b>B</b><td align=center><b>G</b><td align=center><b>R</b><tr><td align=center><b>baseline 1/4s</b><td align=center>1.032<tdalign=center>1.756<td align=center>1.466<td align=center>0.952<tr><td align=center><b>4s + 1/3</b><td align=center>0.013<tdalign=center>0.010<td align=center>0.034<td align=center>0.000<tr><td align=center><b>8s + 1/3</b><td align=center>0.000<tdalign=center>0.019<td align=center>0.024<td align=center>-0.023<tr><td align=center><b>15s +2/3</b><td align=center>0.002<tdalign=center>0.074<td align=center>0.034<td align=center>-0.027<tr><td align=center><b>30s + 2/3</b><td align=center>-0.016<tdalign=center>0.050<td align=center>0.008<td align=center>-0.049<tr><td align=center><b>60s +1</b><td align=center>0.009<tdalign=center>0.146<td align=center>0.049<td align=center>-0.030<tr><td align=center><b>120s +1</b><td align=center>-0.031<tdalign=center>0.146<td align=center>0.003<td align=center>-0.077<tr><td align=center><b>120s +1 1/3</b><td align=center>0.033<tdalign=center>0.213<td align=center>0.084<td align=center>0.002</table></center><br><h4>Conclusion</h4> <p>Both of these films seem eminently usable for moderate durationlong exposures. While the slide films I've used do a bit better, +1stop at 120s is really pretty good. While there are definitely colorshifts, they shouldn't be too hard to correct when printing. <p>Overall, I'd say Reala has the slight edge, since it requiresslightly less exposure compensation and significantly less colorcorrection for very long exposures. Somewhat surprisingly, my resultsshow much less exposure compensation than Fuji's datasheet. I can'treally be sure which is the more accurate number until I experimentwith the film a bit further. <p>I'd appreciate any comments about my testing methods and/or theresults. I'm hoping to test a couple more emulsions in the same way astime permits. Any feedback received will hopefully make subsequenttests more accurate. <p>And of course, the standard disclaimer: I make no promises aboutthe accuracy or validity of this data whatsoever. While I tried toconduct the tests as scientifically as I could given my resources,it's entirely possible I messed up somewhere. I advise you to do yourown testing and treat these numbers as just a starting point. And ifyou do take this data as gospel truth and ruin a shoot because of someerror in them, you only have yourself to blame. <p>-Jon T. Link to comment Share on other sites More sharing options...
maclean Posted February 23, 2005 Share Posted February 23, 2005 I really appreciate you taking the time to do this test because I'm trying to decide between the two films for a night project I'm about to undertake. I have 100UC already and was trying to decide if I need to go buy some Reala instead. I think I will go with the UC and see how it comes out. I'll be shooting storefronts at night in the range of 15-30s exposures. Link to comment Share on other sites More sharing options...
kirk_keyes Posted February 23, 2005 Share Posted February 23, 2005 Jon - really nice job on the testing. I really like that you gave an indication of color balance shifts. My only suggestion for this test, is since you gave us so much info already, would be to state how you calibrated the densitometer (just for the record). For future testing, I'd suggest doing tests of longer times, i.e. say up to 30 minutes. And I understand it is harder to give corrections in time instead of f/stops, but I know I would find that info more useful, as I am usually limited in f/stop and I need to increase time instead of opening the lens up. Also, it would be nice if you could use a light source that you could get better control with, especially if you go to exposure times that are much longer than you did here. And finally, this single expsoure-point type of testing doesn't give any indication as to any contrast changes. Perhaps you could devise a method to test that. Like doing brackets at -2 and +2 stops at each tested time. Any, good write up! Kirk Link to comment Share on other sites More sharing options...
rowland_mowrey Posted February 23, 2005 Share Posted February 23, 2005 Jon; I second the comments here. Very good test. I would like to add something though. Since films often change contrast as a function of reciprocity, it is often good to measure the changes at different spots on the curve by using something like a MacBeth chart and use the step scale there. That way, you see if there is crossover in the image as a function of mismatched contrast reciprociy failure. This can be a serious problem in some color materials. Ron Mowrey Link to comment Share on other sites More sharing options...
bill_tuthill Posted February 23, 2005 Share Posted February 23, 2005 Isn't 100UC's blue-cyan shift more useful for long exposures due to the prevalance of tungsten and sodium lighting at night? Thanks Jonathan, great work. Link to comment Share on other sites More sharing options...
marco_buonocore Posted February 23, 2005 Share Posted February 23, 2005 Thanks Jon, Sounds like a rainy day well spent! I usually just hang around and mope... Anyone interested in your findings might also want to check out: http://home.earthlink.net/~kitathome/LunarLight/moonlight_gallery/technique/reciprocity.htm I've found it to be quite helpful, and I certainly appreciate the effort you lot go into testing film! Cheers, Link to comment Share on other sites More sharing options...
jonathan_traupman Posted February 23, 2005 Author Share Posted February 23, 2005 <p>Thanks for all the comments. I'm glad this info is useful to people. <p>Anyway, as for your questions: <p><b><i>My only suggestion for this test, is since you gave us so much info already, would be to state how you calibrated the densitometer (just for the record).</i></b> <p>The densitometer comes with two reference targets, one for reflective measurements, the other for transmissive. It allows you to enter and store in memory the VBGR densities of the transmissive target and both the high and low densities of the reflective one. Before I measured the film, I hit the "calibrate" button, which then prompts me to take a measurement without any film, then a measurement of the reference film. It then does its own calibration. <p>For this particular test, I let the densitometer warm up for about an hour so that its power supply and lamp were stable, then calibrated, measured the first film, re-calibrated, and measure the second film. <p><b><i>For future testing, I'd suggest doing tests of longer times, i.e. say up to 30 minutes.</i></b> <p>I'd like to do this, too, but can't make any promises. As it is, it took about 45 minutes per film for this test. Making 5-6 test exposures of 30 minutes each will take all day, so I don't know if I have the time. <p><b><i>And I understand it is harder to give corrections in time instead of f/stops, but I know I would find that info more useful, as I am usually limited in f/stop and I need to increase time instead of opening the lens up.</i></b> <p>This is also something I'll hopefully get around to. In the meantime, you can interpolate these charts to get that information. For example, let's say you're shooting UC100 and your meter reads 30s f/5.6. According to the chart, that would require you to open the aperture 2/3 of a stop, but perhaps that's not possible with your lens. So look to the next line, 60s. The metered exposure would be 60s f/8, but the chart requires you to open up one stop, so you shoot at 60s f/5.6. So, in other words UC100 at 30s requires +1 stop of time compensation vs. +2/3 stop of aperture compensation. If the numbers don't match up exactly, you can pretty safely linearly interpolate between times and get close enough. <p><b><i>Also, it would be nice if you could use a light source that you could get better control with, especially if you go to exposure times that are much longer than you did here.</i></b> <p>Yes, the next film I want to look at is Portra 100T, and I'll try to borrow my friend's hot lights to do the test. I need to do a little research to see just how close to daylight tungsten lights with an 80A filter actually is before I use them for testing daylight films. <p><b><i>And finally, this single expsoure-point type of testing doesn't give any indication as to any contrast changes. Perhaps you could devise a method to test that. Like doing brackets at -2 and +2 stops at each tested time. <p>Since films often change contrast as a function of reciprocity, it is often good to measure the changes at different spots on the curve by using something like a MacBeth chart and use the step scale there.</b></i> <p>True. I'll try to get ahold of a grey wedge for my next round of tests. I probably won't be getting a MacBeth chart -- that's one expensive piece of cardboard -- but I have a reflective IT8 target I can use that will at least give a qualitative indication of color shift. I'll have to show scans of the film for that test, since the smallest spot my densitometer can measure is about 2mm diameter. <p><b><i>Isn't 100UC's blue-cyan shift more useful for long exposures due to the prevalance of tungsten and sodium lighting at night?</i></b> <p>Possibly. It all depends on how constant the shift is across the tonal range. I just measured the shift around neutral grey, which can easily be corrected in printing. If the highlights and shadows shift differently than the midtones, it may or may not give you the results you want. <p>My next test is going to be of some tungsten negative film because I've often liked tungsten slide films at night, and tungsten films in general are engineered for longer exposures. <p>Again, thanks for the comments and suggestions. I'll post more results as I get them. <p>--Jon T. Link to comment Share on other sites More sharing options...
rowland_mowrey Posted February 23, 2005 Share Posted February 23, 2005 Jon; Kodak sells a number of gray step scales for reasonable prices. They used to include one in their color dataguide. I know what you mean though. Ron Mowrey Link to comment Share on other sites More sharing options...
kirk_keyes Posted February 23, 2005 Share Posted February 23, 2005 Jon, thanks for the feedback. I understand the amount of time it will take to do longer tests. I've worked out an exposure schedule for using my rarely used 6x7 enlarger to use as a light source for making a series of reciprocity tests. I'm going to use a Stouffer step wedge so that will save some time! But I still need to set aside a chunk of time to actually do it. I've been using my Saunders 4x5 enlarger (w/ halogen bulb) as a light source for film testing, and I was concerned about the color temp. So I rented a color temp meter (only about $20 for the weekend, although I did have to put about $1000 down (on a credit card) to secure it). I think I remember the meter reading 2950K for the enlarger with no filtration, and then I put an 80A on the lens and it read about 5750K. That was close enough for me, and at least I've quanitified it if I need to know it for further work. The IT8 target will be hard to read on a 35mm neg. Since you don't have a large MacBeth chart, I would suggest that you take some photopaper or an inkjet printer and make yourself a reflectance chart. Just print up some tones going from white to black that are several inches in size and then cut them up to whatever size you want your targets to be. You can calibrate them with your Noritsu DM-201. (I have one - I really like being able to read VRGB all with one reading, and both reflection and transmission! Nice meter.) Just make sure you don't get glare on the B&W paper if you make a chart yourself. And don't forget to include a bellow extension factor since you will want to focus the lens on the chart instead of focusing on infinity as before. Kirk Link to comment Share on other sites More sharing options...
bill_tuthill Posted February 23, 2005 Share Posted February 23, 2005 Steve Dunn posted a Macbeth chart PNG for inkjet printing <A HREF="http://www.photo.net/bboard/q-and-a-fetch-msg?msg_id=00AIyW">in this thread</A>. Link to comment Share on other sites More sharing options...
patrick_gainer Posted February 26, 2005 Share Posted February 26, 2005 There still remains a problem for those who must set the aperture for depth of field. It is not easy to translate the f-stop adjustments into time adjustments because opening the lens changes the illumination of the film, which changes the It relationship. In fact, the failure of reciprocity means that strictly speaking there is no longer a direct way to exchange illumination for time. It gets to be a headache. I'm heading for the medicine cabinet now. Link to comment Share on other sites More sharing options...
patrick_gainer Posted February 28, 2005 Share Posted February 28, 2005 Jonathan: This may help or it may not. I was seeking a way to show corrections and indicated exposure in the same units, using your data as base. My venerable Luna Pro meter tells me that your basic exposure of 2.8 at 1/15 second for ISO 100 means that there were an average of 350 meter-candles incident on the scene. Multiplying 350 X 1/15 gives 23.33 mcs exposure, which as far as the meter is concerned applies for any exposure time. I can calculate the "scene" illumination by dividing 23.33 by the indicated exposure time. This gives the following table: time, s Im, meter-candles 4 5.83 8 2.92 15 1.56 30 0.78 60 0.39 120 0.19 Now it is easier to apply the corrections you found. 1/3 f-stop is a factor on illumination of 1.26, 2/3 = 1.58, 1 = 2.0, 1 1/3 = 2.51. I will focus on 30 to 120 seconds because the resolution of f-stops is an appreciable fraction of the corection at less than 1/3 f-stop. For FR100; Time Im Ic Ic*t t' t' - t 30 0.78 0.98 29.4 38 8 60 0.39 0.62 36.9 95 35 120 0.19 0.38 45.6 240 120 t' is Ic*t/Im. For UC100 30 0.78 1.23 36.9 47 17 60 0.39 0.78 46.7 120 60 120 0.19 0.48 58.6 308 188 It is fruitful to plot t' - t against indicated time on log-log paper. You will find that the equation: t = tm + 0.012*(tm^1.92) fits the FR100 data satisfactorilly and t = tm + 0.025*(tm^1.92) fits the UC100 data. These equations read "Corrected exposure time is indicated exposure time plus a factor times indicated exposure time raised to the 1.92 power." The factor is 0.012 for FR100 and 0.025 for UC100. Note that, as far as I can tell from the data, the exponent 1.92 is the same for both films. It is a hypothesis that seems to work, but may be proved wrong by further experiment. These equations should in any case serve as a guide for future tests where varying time instead of f-stop is used to correct for reciprocity failure. Link to comment Share on other sites More sharing options...
patrick_gainer Posted February 28, 2005 Share Posted February 28, 2005 Well, I can see that was a waste of time. How in the world does one get formatting of columns to hold? Link to comment Share on other sites More sharing options...
bill_tuthill Posted February 28, 2005 Share Posted February 28, 2005 <Pre> used to work, but was discontinued for security reasons, maybe buffer overflow. Now I believe <Table> is the only way. Link to comment Share on other sites More sharing options...
patrick_gainer Posted February 28, 2005 Share Posted February 28, 2005 <P>This may help or it may not. I was seeking a way to show corrections and indicated exposure in the same units, using your data as base.</P> <P>My venerable Luna Pro meter tells me that your basic exposure of 2.8 at 1/15 second for ISO 100 means that there were an average of 350 meter-candles incident on the scene. Multiplying 350 X 1/15 gives 23.33 mcs exposure, which as far as the meter is concerned applies for any exposure time. I can calculate the "scene" illumination by dividing 23.33 by the indicated exposure time. This gives the following table:</P> <TABLE BORDER="1" WIDTH="268" CELLPADDING="1" CELLSPACING="1"> <TR VALIGN="TOP"><TD WIDTH="129">Time, seconds</TD> <TD WIDTH="139">Im, meter-candles</TD></TR> <TR VALIGN="TOP"><TD>4</TD> <TD>5.83</TD></TR> <TR VALIGN="TOP"><TD>8</TD> <TD>2.92</TD></TR> <TR VALIGN="TOP"><TD>15</TD> <TD>1.56</TD></TR> <TR VALIGN="TOP"><TD>30</TD> <TD>0.78</TD></TR> <TR VALIGN="TOP"><TD>60</TD> <TD>0.39</TD></TR> <TR VALIGN="TOP"><TD>120</TD> <TD>0.19</TD></TR></TABLE> <P>Now it is easier to apply the corrections you found. 1/3 f-stop is a factor on illumination of 1.26, 2/3 = 1.58, 1 = 2.0, 1 1/3 = 2.51.I will focus on 30 to 120 seconds because the resolution of f-stops is an appreciable fraction of the correction at less than 1/3 f-stop.</P> <P>For FR100;</P> <TABLE BORDER="1" WIDTH="100%" CELLPADDING="1" CELLSPACING="1"> <TR VALIGN="TOP"><TD>Time,t</TD> <TD>Im</TD> <TD>Ic</TD> <TD>Ic*t</TD> <TD>t'</TD> <TD>t' - t</TD></TR> <TR VALIGN="TOP"><TD>30</TD> <TD>0.78</TD> <TD>0.98</TD> <TD>29.4</TD> <TD>38</TD> <TD>8</TD></TR> <TR VALIGN="TOP"><TD>60</TD> <TD>0.39</TD> <TD>0.62</TD> <TD>36.9</TD> <TD>95</TD> <TD>35</TD></TR> <TR VALIGN="TOP"><TD>120</TD> <TD>0.19</TD> <TD>0.38</TD> <TD>45.6</TD> <TD>240</TD> <TD>120</TD></TR></TABLE> <P>t' is Ic*t/Im. </P> <P>For UC100</P> <TABLE BORDER="1" WIDTH="100%" CELLPADDING="1" CELLSPACING="1"> <TR VALIGN="TOP"><TD>Time,t</TD> <TD>Im</TD> <TD>Ic</TD> <TD>Ic*t</TD> <TD>t'</TD> <TD>t' - t</TD></TR> <TR VALIGN="TOP"><TD>30</TD> <TD>0.78</TD> <TD>1.23</TD> <TD>36.9</TD> <TD>47</TD> <TD>17</TD></TR> <TR VALIGN="TOP"><TD>60</TD> <TD>0.39</TD> <TD>0.78</TD> <TD>45.7</TD> <TD>120</TD> <TD>60</TD></TR> <TR VALIGN="TOP"><TD>120</TD> <TD>0.19</TD> <TD>0.48</TD> <TD>58.6</TD> <TD>308</TD> <TD>188</TD></TR></TABLE> <P>It is fruitful to plot t' - t against indicated time on log-log paper. You will find that the equation:</P> <P>t' = tm + 0.028*(tm^1.73)</P> <P>fits the FR100 data satisfactorily and</P> <P>t' = tm + 0.055*(tm^1.73)</P> <P>fits the UC100 data. These equations read "Corrected exposure time is indicated exposure time plus a factor times indicated exposure time raised to the 1.92 power." The factor is 0.012 for FR100 and 0.025 for UC100. Note that, as far as I can tell from the data, the exponent 1.92 is the same for both films. It is a hypothesis that seems to work, but may be proved wrong by further experiment. These equations should in any case serve as a guide for future tests where varying time instead of f-stop is used to correct for reciprocity failure. </P> </BODY> </HTML> Link to comment Share on other sites More sharing options...
ryan_buckley Posted September 13, 2008 Share Posted September 13, 2008 using these equations, when plugging in the metered time, does it need to be in seconds? for example if i know the tm is 30mins: t' = 30 + 0.055*(30^1.73) = 50 minutes whereas, in seconds it's 423 minutes. Link to comment Share on other sites More sharing options...
m._b._hardy Posted May 1, 2009 Share Posted May 1, 2009 <p>If derived for seconds, it must stay in seconds, but it can be converted. Failure to use the correct units conversion with a power law can cause gross error. A constant multiplier, F^(p-1), can be added to the right term of the power law, Tc=To+a*^p, for non-standard units of time, where F is the conversion factor between units. If a and p are derived for Tc, To, and Tm in seconds, the right term must be multiplied by 60^(p-1) for Tc, To, and Tm in minutes.<br /> <br /> Derivation with F=60: For Tc, To, and Tm in minutes, write them in seconds, 60*Tc=60*To+a*(60*Tm)^p=60*To+a*60^p*^p. Divide both sides by 60 and commute a to get Tc=To+60^(p-1)*a*^p for Tc, To, and Tm in minutes with the correct conversion factor. In the case above, 60^(1.73-1)=19.8632548.</p> Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now