Calculation for cross section and area efficiency of progressing cavity pump with hypotrochoidal multilobe using the differential geometric envelope approach

Now progressing cavity pumps (PCPs) and downhole mud-motors (DHMs) with cycloidal profiles are widely used in oil and gas industry. This study presents the method to design the cross section of PCP or DHM with hypotrochoidal multilobe. The conditions for conjugate envelope of hypotrochoidal rotor were derived by using the differential geometric approach. On the basis of this, we made the formulae for the envelope and its offset curve and worked to calculate the area efficiency of cross section and curvature of hypotrochoid. The field of trochoid ratio and offset distance were selected to get the rational offset curve. These results enable to design the cross section of PCP or DHM with multilobe of hypotrochoid which has the continuous and limited curvature. These results also can be used for multilobe PCP or DHM with solid stator.