We study the robustness of flow networks against cascading failures under a partial load redistribution model. In particular, we consider a flow network of N lines with initial loads L_1,... , L_N and free-spaces S_1,... , S_N that are independent and identically distributed with joint distribution P_{LS}(x,y). The capacity C_i is the maximum load allowed on line i, and is  given by C_i=L_i + S_i. When a line fails due to overloading, i.e., due to its load exceeding its capacity, 
it is removed from the system and (1-epsilon)-fraction of the load it was carrying (at the moment of failing) gets redistributed equally among all remaining lines in the system; hence we refer to this as the {em partial} load redistribution model. The rest (i.e., epsilon fraction) of the load is assumed to be lost or absorbed, e.g., due to advanced circuitry disconnecting overloaded power lines or an inter-connected network/material absorbing a fraction of the flow from overloaded lines. We analyze the robustness of this flow network against random attacks that remove a p-fraction of the lines. Our contributions include (i) deriving the final fraction of alive lines n_{infinity}(p,epsilon) for all p, epsilon in (0,1) interval and confirming the results via extensive simulations; (ii) showing that partial redistribution might lead to (depending on the parameter 0<epsilon < 1)  the order of transition at the critical attack size p* changing from first to second-order; and (iii) proving analytically that the widely used robustness metric measuring the area under n_{infinity}(p,epsilon) is maximized when all lines have the same free-space regardless of their initial load.