Recent results have shown that interference alignment can achieve K/2 degrees of freedom in a K-user interference channel with time or frequency varying channel coefficients. For fixed number of users K, the number of degrees of freedom characterizes the asymptotic behavior of the performance in the high SNR limit but it does not answer the question of how the performance scales with K for any fixed SNR. In particular, it is unclear if a constant rate per user can be maintained as more users enter into the system. In this paper, we investigate the performance of the interference alignment scheme for fixed SNR. We assume that the channel coefficients between the users are of the form r e^{j\theta} where r is fixed over the duration of communication and theta is a fast fading phase. We show that for any value of the SNR and K, the aggregate rate achieved by the interference alignment scheme is lower bounded by c_1 K log(1+c_2 SNR) where c_1 and c_2 are positive constants independent of both SNR and K. This result establishes the linear scaling of the interference alignment scheme for the considered random phase channel model.