In potential flow networks, the equilibrium flow rates are usually not proportional to the demands and flow control elements are required to regulate the flow. The control elements can broadly be classified into two typesโdiscrete and continuous. Discrete control elements can have only two operational states: fully open or fully closed. On the other hand, continuous control elements may be operated in any intermediate position in addition to the fully open and fully closed states. Naturally, with their increased flexibility, continuous control elements can provide better network performance, but to what extent? We consider a class of branched networks with a single source and multiple sinks. The potential drop across edges (๐ฅ๐ป) is assumed to be proportional to the ๐th power of flow rate (๐), i.e., ๐ฅ๐ป = ๐๐๐ , (๐ โฅ 1). We define R as the ratio of minimal operational times required to transport a given quantum of material with either type of control element and show that 1 โค R โค ๐(1โ1โ๐) , where ๐ is the maximum depth of the network. The results point to the role of network topology in the variations in operational time. Further analysis reveals that the selfish operation of a network with continuous control valves has the same bounds on the price of anarchy.