1. Formation of kidney stones occurs when concentration of a poorly soluble salt (for example, calcium oxalate) in urine exceeds its solubility limit. Assume that a kidney stone has already formed and has traveled into the bladder. A bioengineer hypothesizes that the stone can be dissolved in the urine and excreted. The stone weighs 10 grams and is spherical in shape (density of 3 gm/cm3). The rate of stone dissolution is given by the following equation:
Mlost =kA(Csat -C )
where, Mlost is the amount of stone lost per unit time, Csat is the saturation concentration of the salt in the urine (1mg/ml), C is the salt concentration in the urine. A is the stone surface area andk is the mass transfer coefficient (0.001 cm/s). Assume that the bladder is a well-mixed compartment whose average volume is 400 ml and is emptied every 4 hours. How long will it take for the stone to lose 99% of its mass? To facilitate the removal, lithotripsy is performed. In this procedure, high intensity ultrasound is applied to the stone to break it into n spherical stones. What should be the value of n so that the removal time is reduced by a factor of 5?