The courtroom drama pitting Gina Rinehart, Australia's richest person, and her estranged children will continue playing out before the public as the New South Wales Court of Appeal threw out on Friday her plea for private arbitration of the case.

Ms Rinehart has been waging a legal battle to keep control of the $2.4-billion Hope Margaret Hancock trust, which was set up in the late 1980s by Hancock Prospecting founder Lang Hancock, Rinehart's late father, to benefit his four grandchildren namely John Hancock, Hope Welker, Bianca Rinehart and Ginia Rinehart.

Media reports have estimated that the trust fund holds some 23 percent ownership of Hancock Prospecting, the holdings company that manages Rinehart's iron ore projects in Australia.

Rinehart's mining interests have been the prime generators of her vast fortunes, which this year were estimated by U.S.-based Fortune Magazine at about $18 billion, making her the top billionaire in the country and number 29 in the world.

Yet the Rineharts' attracted public attention not only due their immense wealth but also because of the series of family intrigues that have been plaguing their relationships, the latest of which is the legal stoush that would determine stewardship of the billions of dollars that the Hancock patriarch had left behind, apparently for his grandchildren's taking.

That would have become a reality last year when Ginia turned 25 last year but Rinehart petitioned the court to keep her as trustee of the fund, arguing that her children were not prepared to handle the serious responsibilities entailed in managing huge amount of money.

Rinehart also informed the court that her actions were motivated by intents to encourage her children to work and become productive while at the same time requesting that the details of the dispute be kept out of public access.

Her request was rebuffed, however, earlier this year by the Australian High Court, which was followed today by the decision of NSW Court of Appeal Chief Justice Tom Bathurst.

"I have concluded that the respondents' claim was not a dispute under the settlement deed, that the settlement deed did not have the effect of barring any claim for removal of the trustee and that the other defences, although arising under the settlement deed, did not necessarily bar a claim for removal of GHR (Gina Rinehart) as trustee," Chief Justice Bathurst was reported by The Australian as saying in his ruling.

The same court ruling also ordered Rinehart to shoulder the payments of the legal costs that the dispute has so far caused.

With arbitration now out of the question, the Supreme Court will now handle the final settlement of the case, which is set to take place later this year.

The legal spar within the Rinehart family has highlighted the fissures between Gina Rinehart and her children, with only Ginia deciding to throw her lot with mother.