The constrained maximum flow problem is a variant of the classical maximum flow problem in which the flow from a source node to a sink node is maximized in a directed capacitated network with arc costs subject to the constraint that the total cost of flow should be within a budget. It is important to study this problem because it has important applications, such as in logistics, telecommunications and computer networks; and because it is related to variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications as well. In this research, we present an O(n^2mlog(nC)) time cost scaling algorithm and compare its empirical performance against the two existing polynomial combinatorial algorithms for the problem: the capacity scaling and the double scaling algorithms. We show that the cost scaling algorithm is on average 25 times faster than the double scaling algorithm, and 32 times faster than the capacity scaling algorithm.