Industrial process monitoring increasingly relies on sensor-generated time-series data, yet the lack of labels, high variability, and operational noise make it difficult to extract meaningful patterns using conventional methods. Existing clustering techniques either rely on fixed distance metrics or deep models designed for static data, limiting their ability to handle dynamic, unstructured industrial sequences. Addressing this gap, this paper proposes a novel framework for unsupervised discovery of operational modes in univariate time-series data using image-based convolutional clustering with composite internal evaluation. The proposed framework improves upon existing approaches in three ways: (1) raw time-series sequences are transformed into grayscale matrix representations via overlapping sliding windows, allowing effective feature extraction using a deep convolutional autoencoder; (2) the framework integrates both soft and hard clustering outputs and refines the selection through a two-stage strategy; and (3) clustering performance is objectively evaluated by a newly developed composite score, S_eva, which combines normalized Silhouette, Calinski-Harabasz, and Davies-Bouldin indices. Applied to over 3900 furnace melting operations from a Nordic foundry, the method identifies seven explainable operational patterns, revealing significant differences in energy consumption, thermal dynamics, and production duration. Compared to classical and deep clustering baselines, the proposed approach achieves superior overall performance, greater robustness, and domain-aligned explainability. The framework addresses key challenges in unsupervised time-series analysis, such as sequence irregularity, overlapping modes, and metric inconsistency, and provides a generalizable solution for data-driven diagnostics and energy optimization in industrial systems.