An introduction to temperature and dissolved oxygen in Hydroponics:
In order to maximize plant growth hydroponically it’s necessary to balance a number of factors. Sometimes these factors can be interrelated, as is the case for the temperature of a nutrient solution and the capacity of that nutrient solution for dissolved oxygen. For this reason it's necessary to maintain a nutrient solution that is neither too cold nor too warm. For most applications the desired temperature will fall within a roughly defined range of 18-24 °C (54-75 °F). This article is intended to summarize and introduce the relationship between a nutrient solution's temperature and capacity for dissolved oxygen (DO) in a way that is both straight forward and relevant to hydroponic gardeners while preparing them for more technically related discussions regarding the subject.  

Water Chemistry:
To begin, it's important to understand that water can absorb oxygen from the atmosphere. Depending on water temperature, salinity and air pressure, the extent to which this occurs can range anywhere from 0 parts per million (ppm) to 15 ppm.  Keep in mind that at this point we are referring only to the hypothetical maximum DO level. If dissolved oxygen within a nutrient solution is being used more quickly than it can be replenished from the atmosphere then the actual DO level would be less than this hypothetical maximum, assuming a maximum greater than zero. For all practical purposes a hydroponic gardener needs to only to be concerned with the role of temperature. Simply, as water cools the potential maximum DO level increases and conversely as water warms the maximum DO level decreases.

Although many websites and books discussing the relationship between DO saturation and temperature for hobbyist hydroponic gardening have thrown around seemingly made up numbers while providing zero references, Bilberrybrian.com believes in doing it better. The graph below illustrates the relationship between temperature and DO saturation while highlighting the approximate zone optimal for hydroponic gardening. If you wish to plot this graph yourself you may do so using the formula:

14.652 + 0.41022T + 0.0079910T² + 0.000077774T³

This formula was taken from page 1.75 of the McGraw-Hill Handbook of Environmental Engineering Calculations 2^nd Ed. The calculation performed here assumes standard air pressure and pure water. 

Figure 1: Maximum possible DO concentrations for fresh water at sea level. 

Plant Physiology:
Since plants require oxygen not only at the crown level (everything above ground) but also in the root zone, the availability of oxygen in a nutrient solution is a detail that should not be overlooked by the hydroponic gardener. Unfortunately, optimal growing conditions cannot be achieved simply by maintaining an icy cold nutrient solution, thereby increasing the maximum possible DO level. At any moment a growing plant is performing a countless number of biochemical reactions. The “quickness” that these reactions can take place is dictated largely by the temperature of the surrounding environment. As a general rule, the rate of these biochemical reactions will double given for every increase of 10 °C degrees. It would be more appropriate to say in this case that the rate of a plant’s biochemical reaction will be cut in half for every 10 °C decrease. If this all seems contradictory don’t let that frustrate you. All that's necessary is a simple compromise between temperatures that are too cold and too warm.

What several authorities advise:
Dr, Resh, author of Hydroponic Food Production states in his book that a nutrient solution should generally be maintained between 16-18 °C (60-65 °F). During a personal correspondence Harley Smith, professional hydroponic consultant and research director of Hydrodynamics Intl., recommended 20-22  °C (68-72 °F). The now defunct Integral Hydroponics by G. Low recommended 22-23 °C (68-73 °F). George Van Patten recommends to the readers of Gardening Indoors that they maintain a nutrient solution of 16-24 °C (60-75 °C). Taken as a consensus from these and additional sources it can safely be stated that an ideal temperature is approxiamtely 18-24 °C (54-75 °F). These temperatures also apply while using aeroponic  methods . Please  note that this information should not serve as guidelines during seed germination. The above information is repeated below in table 1.

Table 1: Recommended temperatures for a nutrient solution

Summary:
In conclusion, hydroponic gardeners should generally maintain their nutrient solution between 18-24 °C (54-75 °F) since plant productivity can become adversely affected when the nutrient solution is either excessively warm or cold. This applies only specifically during mature growth. Depending on the species in question, seedling germination is often favored given warmer temperatures of 20-30 °C (68-86 °F). Additionally, steps should be taken to ensure that adequate water aeration and turnover is provided using air stones and/or water pumps. Although beyond the scope of this article, several methods and instruments exist for the purpose of testing the oxygen concentration of a nutrient solutions. Because this information has been presented as a general overview and introduction, details regarding specific complications involving plant pathology (disease), diminished nutrient uptake, etc. have been omitted. 

References and Additional Reading:

C. Lee. Handbook of Environmental Engineering Calculations. p. 1.75
American Society of Civil Engineering Committee on Sanitary Engineering Research, 1960.

Fred Senese. How can I predict the oxygen solubility in water? 2010. http://antoine.frostburg.edu/chem/senese/101/solutions/faq/predicting-DO.shtml (Accessed February 7, 2010)

Fred Senese.  Does a 10°C temperature rise double reaction rates? 2010. http://antoine.frostburg.edu/chem/senese/101/kinetics/faq/temperature-and-reaction-rate.shtml  (Accessed February 7, 2010)

Reyes, D. M, Stolzy, L. H., Labanauskas, C. K. Agron J. 1977. Temperature and Oxygen Effects in Soil on Nutrient Uptake in Jojoba seedlings. http://agron.scijournals.org/cgi/content/abstract/69/4/647 (Accessed February 7, 2010)

Selmer-Olsen, A.R., Gislerød, H.R. 1982 Effect of Root Temperature on Nutrient Uptake by the Chrysanthemum. http://www.actahort.org/books/126/126_48.htm (Accessed February 7, 2010)

Setter, T.L., Greenway, H. 1988. Growth Reductions of Rice at Low Temperature: Decreases in Nutrient Uptake and Development of Chlorosis. http://jxb.oxfordjournals.org/cgi/content/abstract/39/6/811?ck=nck (Accessed February 7, 2010)

Heirloom Seed.  Germination tables from Heirloom Seeds - Know when to plant your vegetables. http://www.heirloomseeds.com/germination.html (Accessed February 7, 2010)