In "my last duchess", why was the duke unhappy with his wife?

It occurs to me that there were several things the Duke could have done with his young wife if he decided he didn't want her. He was powerful enough to have gotten a divorce, even from the Catholic Church in those days. If not a divorce, he could have gotten an annulment. He could have simply arranged a separation, with his wife living in a distant place. He could have sent her back to live with her parents. Or, as Browning himself suggested, he could have had her confined to a convent. But the reason he had her killed was very likely to be able to marry again and collect another dowry. That is the main reason he has invited his visitor, the intended bride's father's representative, upstairs for a private talk.