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A novel design of an air gap membrane distillation (AGMD) module was proposed to enhance the permeate flux improvement for the desalination of pure water productivity. The modeling equations for predicting permeate flux in the AGMD module by inserting $\mathsf{\Lambda }$-ribs carbon-fiber open slots under various hydrodynamic angles were developed theoretically and experimentally. The temperature distributions of both hot and cold feed streams were represented graphically with the hot saline flow rate, inlet saline temperature, and carbon-fiber hydrodynamic angles as parameters. The results showed a good agreement between the experimental results and theoretical predictions. Designed by inserting $\mathsf{\Lambda }$-ribs carbon-fiber open slots into the flow channel, the membrane distillation module was implemented to act as an eddy promoter and yield an augmented turbulence flow. The effect of $\mathsf{\Lambda }$-ribs carbon-fiber open slots not only assured the membrane stability by preventing vibration but also increased the permeate flux by diminishing the temperature polarization of the thermal boundary layer. The permeate flux improvement by inserting $\mathsf{\Lambda }$-ribs carbon-fiber open slots in the AGMD module provided the maximum relative increment of up to 15.6% due to the diminution of the concentration polarization effect. The experimental data was incorporated with the hydrodynamic angle of $\mathsf{\Lambda }$-ribs carbon-fiber open slots to correlate the enhancement factor with the Nusselt numbers to confirm the theoretical predictions. The accuracy derivation between the experimental results and theoretical predictions was pretty good, within $9.95\le E\le 1.85$. The effects of operating and designing parameters of hot saline flow rate, inlet saline temperature, and hydrodynamic angle on the permeate flux were also delineated by considering both the power consumption increment and permeate flux enhancement. Full article