Concept:
In industrial chemistry, specific named chemical processes are standardized for the mass production of essential commodity chemicals. Matching the product to its historical manufacturing process is a fundamental chemical engineering concept.
Step 1:
The question asks for the manufacturing process of "Soda ash", which is the common name for sodium carbonate ($Na_2CO_3$).
Step 2:
The Contact process is universally known for the industrial production of Sulfuric Acid ($H_2SO_4$), not soda ash.
Step 3:
The Haber (or Haber-Bosch) process is used to synthesize Ammonia ($NH_3$) from nitrogen and hydrogen gases.
Step 4:
The Ostwald process is the standard chemical process for manufacturing Nitric Acid ($HNO_3$) by oxidizing ammonia.
Step 5:
The Solvay process uses brine (sodium chloride), limestone (calcium carbonate), and ammonia to successfully produce Soda Ash (sodium carbonate), making it the correct answer.