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β-ᴅ-glucose is more stable than its α-anomer due to the fact that all the −OH groups occupy the equatorial position. If this is the case, then why does α-ᴅ-glucose predominate at 30 °C while β-ᴅ-glucose does the same at 98 °C?

Here is an excerpt from PubChem's page for D(+)-Glucose :

Glucose is a reducing sugar, i.e. it reacts with oxidizing agents such as cupric hydroxide ... In water, ... below a temperature of ca 100 °C, the stable, crystalline form of D-glucose is the alpha-form, which crystallizes as a monohydrate below 50 °C, above 100 °C, the beta-anhydrous form, is most stable (Ref. 1).

Below 50 °C, alpha-d-glucose hydrate, above 50 °C anhydrous form is obtained and at higher temp beta-d-glucose is formed (Ref. 2).

References

  1. Schenck FW; Glucose and Glucose-Containing Syrups. Ullmann's Encyclopedia of Industrial Chemistry. 7th ed. (1999-2016). New York, NY: John Wiley & Sons. Online Posting Date: 15 Dec 2006

  2. O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 824

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β-ᴅ-glucose is more stable than its α-anomer due to the fact that all the −OH groups occupy the equatorial position.

This is specifically in aqueous solution. In the solid, you also have to consider the intermolecular interactions between glucose molecules.

If this is the case, then why does α-ᴅ-glucose predominate at 30 °C while β-ᴅ-glucose does the same at 98 °C?

At both temperatures, there is a mixture of the two in aqueous solution. I could not find how the equilibrium constant changes with temperature.

Why does α-ᴅ-glucose form at a lower temperature than β-ᴅ-glucose?

When you crystallize glucose, the least soluble solid will form predominantly (under thermodynamic rather than kinetic control). Solubilities are temperature-dependent. This property is used in the preparation of the solid alpha form (as a monohydrate) and the beta form (anhydrous):

The crystalization and purification of β-D-glucose from a mixture containing α and β anomers of the sugar illustrates how different reaction conditions can change the outcome of a process, for example, through changes in the relative solubilities of different anomers at different temperatures. In most commercial preparations of crystalline D-glucose, the α anomer predominates because it is less soluble at room temperature. However, at higher temperatures α-D-glucose becomes considerably more soluble and it is the β-D-glucose that is selectively crystallized as it becomes dissociated from the polar solvent.

Source: Reyes-de-Corcuera, J. I.; Teruel, M. A.; Jenkins, D. M. Crystallization of β-ᴅ-Glucose and Analysis with a Simple Glucose Biosensor. J. Chem. Educ. 2009, 86 (8), 959. DOI: 10.1021/ed086p959.

Counter-intuitively, you purify these crystals by recrystallization at low temperature (where the alpha form is less soluble). This works because the sample contains much more beta form than alpha form, so the beta form will crystallize. The low temperature prevents fast mutarotation in the aqueous state. The procedure is described below:

One hundred grams of the sugar are poured into 100 cc. of water at $0^\circ$. The $\beta$ form of glucose is very soluble and with vigorous stirring all dissolves in a few seconds. This solution is quickly filtered and 500 cc. of absolute alcohol are mixed with the filtrate. Seeding and stirring causes an immediate and rapid crystallization of $\beta$-glucose. By this method $\beta$-glucose has been recrystallized several times in succession and the specific rotation determined after each crystallization, the rotation being observed within one minute of the time when the sugar was dissolved in water at $0^\circ$. \begin{array}{} &\text{First experiment} &\text{Second experiment}\\ \mathrm{Original\,\beta - glucose} &+24.5^\circ & + 27^\circ \\ \text{After first recrystallization} &+ 21^\circ & + 21^\circ \\ \text{After second recrystallization} &+ 19.8^\circ & + 19.8^\circ \\ \text{After third recrystallization} &+ 19,5^\circ & + 19.9^\circ \\ \text{After fourth recrystallization} &+ 19.7^\circ & + 19.2^\circ \\ \end{array}

It appears that after the second recrystallization the $\beta$-glucose is free from the $\alpha$-isomer. In each of these experiments the initial specific rotation of the $\beta$-glucose which resulted from the fourth recrystallization has been determined by observing the progress of the mutarotation in water at $0.7^\circ$, where the rate is slow, and extrapolating the vales to the time of dissolving the sugar. By this method the initial specific rotation of $\beta$-glucose has been found to be, in the first experiment $+19.2^\circ$ and in the second one $+18.8^\circ$, giving an average of $+19.0^\circ$. This value agrees completely with that which C.Tanret records in his lates measurement. Roux found it to be $+19.8^\circ$.

Source: Hudson, C. S.; Dale, J. K. Studies on the forms of ᴅ-glucose and their mutarotation. J. Am. Chem. Soc. 1917, 39 (2), 320–328. DOI: 10.1021/ja02247a017.

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