Developer Temperature Compensation Formula

[gainer]

I'm going to make a comment or two about terminology, and the I hope I can worry about something else.

An example of an equation is:

E=MC^2

The only constant specified in that equation is th power of 2. It has served us well as a model to use to search for ways to prov or disprove the basic relationship it represents. We use many such equations in photography which we hope to be true for all material relationships. Most of them we know to be working relationships that do not always work, as for instance the reciprocal relationship between length and illumination of film exposure for constant density, so we have tables of reciprocity corrections. These "corrections" are also subject to description by equation, which may or may not always work. I will say once again: I did not use any basic equations that were not well known when I started this project several years ago. All the equations I used are to be found in "Principles of Optics" by Hardy & Perrin. The ony things I added were a way to estimate gamma infinity by successive approximation from two Gamma-time pairs from the same film-developer combination at the same temperature, which times were not necessarilly in the ratio 1:2, and to point out a transformation that allows easy and informative plotting on semi-log graph paper. This is a lossless, reversible transformation. The assumptions of linearity were not mine, but they were obvious from the equations presented by Hardy & Perrin. I made no illogical mathematical statements. My purpose was not to further the basic knowledge of photographic science but to show it in a different form that made it much easier to extrapolate and interpolate existing data.

 

[gainer]

Now I worry about typos.

 

[Kirk Keyes]

As for curve fitting software, all I can say is that all EK software for reading densities automatically fit them to the cubic spline that was derived from the mathematics gurus work.

That would be fun to have. I suppose we'd need a mainframe to run it thought...

 

[Photo Engineer]

The densitometery equipment was attached to DEC PDP-11 machines that required about 3 or 4 cabinets and racks that reached from floor to celining. The densitometers all were automated and the film or paper was placed into holders with a code notch. The exposures were read and plotted and the data was returned in 24 hours or less to the originator as either a data tape or disk or paper plots in either B&W or color depending on product.

The curves could then be compared with aims for that product.

PE

 

[ic-racer]

1. Develop to a given straight line slope and draw a line along the straight line to intersect the x axis. This log E value is then the "speed" and you can arbitrarily assign it based on the ISO you used in exposure.

Excellent!

 

[Photo Engineer]

That would be fun to have. I suppose we'd need a mainframe to run it thought...

After more thought, I thought that you would have more to say about the kinetic and thermodynamic implications of this discussion.

I hope I have not taken your name in vain on the thread about sharpness.

PE

 

[Kirk Keyes]

Post made in the other thread...

I did get a deva vue reading over this thread of threads of years past that we've all enjoyed ;^).

Thanks to all - I get bored with pink T-max threads all the time.

 

[gainer]

Just remember that it is not the characteristic curve I am fitting, but the (usually) first order reaction curve that is the same basic curve as the change in potential with time of an ideal capacitor being charged from 0 to a constant source through an ideal resistor. That curve seems to fit quite well the tables supplied by Kodak in publication J-109.

 

[Photo Engineer]

Patrick, I think this changes some of my perception of what you have been saying. The first order is the slope of the characteristic curve.

PE

 

[gainer]

I don't yet know if we are talking about the same thing. The basic equation relating a gradient G at one temperature to the maximum gradient is assumed to be:

G = Gmax[(1 - e^(-kt)]

where e is the base of the natural logarithms. When we divide both sides by Gmax , we can plot the resulting equation on semi-log graph paper as Gmax - G on the log axis vs t on the linear axis. Since Gmax has been found to be independent of temperature, the constant k is the only other number we must know in order to define the value of G for any development time t at this temperature. If anyone is interested, I can provide more detail on the calculations, which includes how to generalize to a wide ranges of temperatures.

Certainly each film-developer combination has its own pecular characteristic curve shape, which makes automating the tabulating of development charts a more complicated business, but once the gradients are known, the development time-temperature relationships are much more amenable to generalization in form, which allows a certain degree of reduction of experiments neded for any one film-developer combination. That is not such a big deal for Kodal or Ilford, but for you and me the notion of being able to reduce the number of characteristic curves for one film by a factor of at least 4 is enticing.

 

[gainer]

I should have defined t as time. It does get confusing. The constant k is where temperature enters in.

 

[Photo Engineer]

Well, we define gradient or slope of a curve at any point and time of development as dy/dx where y = density and x = Log E. That gives slope or contrast. Thus the curve of the print can be gotten from the curve of the negative and the curve of the print material by this: dy(p)/dx(p) = dy(f)/dx(f) * dy(m)/dx(m), where p = print, f = film and m = print material or paper. So, by doing this point by point or using the equation which I described before as a spline, you can take two curves and get the result when printing. This assumes no flare, and equivalent neutral density at each point from 400 - 700 nm. The first can be produced in the lab easily, and the second is assumed as any slight error is tiny. Silver itself is never exactly neutral as we all know.

PE

 

[gainer]

I should have said simply that the relationship between time of development and the gradient or CI of a particular film-developer combination is a first order differential equation that can be defined by two constants. When the equation is arranged so that it can be plotted as a straight line on semi-log paper, the intercept is Gmax (or gamma infinity) and the slope of the line is k in the equation I posted above. This equation does not "know" how the values of G were measured or defined. Another way to define a first order system is that the rate of reaction at any time is proporional to the amount of material that remains to be acted upon. The solution of this equation represents experimental curves as satisfactorily as any equation containing only two arbitrary constants, and is quite precise enough, and its greater simplicity justifies its use, according to Hardy & Perrin.

 

[Photo Engineer]

Basically then, as I have contended, each film and developer combination has a different slope as the constants differ. This is from the laws of Kinetics and Thermodynamics cited above.

PE

 

[gainer]

Basically then, as I have contended, each film and developer combination has a different slope as the constants differ. This is from the laws of Kinetics and Thermodynamics cited above.

PE

Yes, indeed. And the family of data that represents the gradient-time-temperature variation for any one of these film-developer combinations can be recreated satisfactorily by three properly chosen points from those data.

 

[Photo Engineer]

So, we agree totally after all of this, and essentially disagree with the OP then. (I think)

PE

 

[ic-racer]

The basic equation relating a gradient G at one temperature to the maximum gradient is assumed to be:

G = Gmax[(1 - e^(-kt)]

Capacitor charging, exponential growth, Newtonian cooling and now film/temp compensation.

As a practical measure, how might one obtain Gmax? Develop at like 75c or something?

 

[gainer]

I think Gmax, or as it is often called, gamma infinity, is most easily obtained by estimating it from two gradients measured at the same temperature. It will be advantageous to have semi-log graph paper at hand. I found a web site from which one can download pdf files which allow one to print out on demand a number of different special graph papers. Use http://incompetech.com/graphpaper/logarithmic/.

Start with an estimate of Gmax. I can be pretty wild. I use 2.0 in my program. Using this value and two known values of CI at the same temperature, calculate 2.0 - CI for each CI, plot these values against development time (time on the linear axis), draw the line connecting these two points and use the value where it intersects the log axis as a new estimate of Gmax. Use that value to calculate Gmax - CI for each point, plot a new line, and repeat this process until two consecutive values of Gmax are close enough for government work.

 

[ic-racer]

What I was trying to do is see if I can get "k" for each of my films. I already have boatloads of "G at temp X" data. If I had G-Max, then I could solve for "k". Then I would be set. After years of computer use, I don't think I know how to use graph paper and pencil anymore I'm going to have to re-read you response a few times to get it.

 

[ic-racer]

Now that I have digested you response I see I can use two curves at the same time but different temperatures (two G's) to obtain "k" mathematically, rather than graphing.

I believe the following is true of the exponential equation listed previously:

k = ln (1+ (r/100))

Where:
k = constant from the exponential equation I am seeking
r = percent increase in G per unit temperature


So, once I have 'k' from the above relationship, I believe I can solve for G-max.

This would really be great if it works.

 

[gainer]

Now that I have digested you response I see I can use two curves at the same time but different temperatures (two G's) to obtain "k" mathematically, rather than graphing.

I believe the following is true of the exponential equation listed previously:

k = ln (1+ (r/100))

Where:
k = constant from the exponential equation I am seeking
r = percent increase in G per unit temperature


So, once I have 'k' from the above relationship, I believe I can solve for G-max.

This would really be great if it works.

I have a feeling that it will not work. The constant k is not constant with temperature, but Gmax is. There appears to be no slope information when constant temperature samples are used. I won't tell you not to try. I could be wrong. There's a first time for everything. Let me give you some data points for which I know the answer and see what you can come up with. G1= .55, t1 = 20 C; G2 = .65, t2 = 24 C. Time was 12 minutes for both. You should get a Gmax = .86, give or take a little for reading error. I have numerous other data points to check with if you can come up with a way to predict them.

 

[Photo Engineer]

Patrick;

For the numbers you give, based on what we have discussed, you have to give a film and developer to match the data to. I too have reservations about the derivation of our fellow APUG member. I read it earlier and something bothered me so I did not post, but then you are better at math than I am I think.

Patrick, I truly wish we could meet or talk sometime. We have so much information to exchange.

PE

 

[ic-racer]

Maybe I have misread something?? Is the little 't' in that original equation Temp or time??

I thought you had posted the formula to relate to temp. But I am thinking (by the way you are writing) that I have misinterpreted things. You are just relating any G to the G-max for given TIMES right. I thought TEMP.

 

[Photo Engineer]

Check with Patrick, but my read on this is that it only works for one film and one developer. You have to get new constants for each combination.

PE

 

[gainer]

The little t is for time. The k that goes with it is variable with temperature as well as the film-developer combination. Since my last post, I have found that it is theoretically not possible to define experimentally all the constants of that equation for even one film-developer combo without measuring some variations due to both time and temperature. Going back to the differential equation:

dG/dt = k( Gmax - G)

If indeed you measure a small increment of each of G and t at a known value of G you can get pretty close to dG/dt and if you do it for 2 values of G you will have a simple simultaneous set to solve. BUT the values you must measure are very much smaller than Gmax which makes the required precision of measurement much to high for practicality. You still must have at least enough information about variation with time to get the derivatives you need.

OTOH, you only need to do two characteristic curves at different development times for each film of interest to be able to choose your time and temperature to get a desired CI for any one film-developer combo in your stable. You already have the temperature data reduced.