The concept of Urban Air Mobility (UAM) services was created mainly in response to traffic congestions. In this research we focus on UAM services such as those provided by Uber Elevate. We therefore present a framework to solve the Urban Air Mobility Problem with Time Windows (UAMP-TW) under dynamic demand, using an Adaptive Large Neighborhood Search (ALNS) algorithm. The objective of this study is to maximize the operational profit and consider customer satisfaction. Satisfaction is measured by two factors: (1) deviation from desired departure time to actual departure time and (2) deviation from nominal trip duration to actual trip duration. In our analysis we aim to determine a relationship between customers and their contribution towards profit. We address this by running simulation instances that cover three operational scenarios: a morning and evening commuter transportation case (scenarios 1 and 2) and the an occurrence of an event at a specific location (scenario 3). Multiple simulation runs indicated stability, for all three instances, due to low variation of the profit from the mean. A sensitivity analysis on the customers' time-window lengths, satisfaction factors and types concluded that customers with higher time-window lengths are more profitable since it is easier to share-rides with other users. The analysis also showed that when the satisfaction factors have a higher weight in the deviation from the departure time than the trip duration, the overall customer satisfaction is increased together with the profit and the percentage of customers who share rides. Scenario 1 has a higher rate of rebalancing empty vehicles because most requests are generated in the suburbs while the depot is located downtown. This leads to a lower vehicle deployment. In scenarios 2 and 3, most requests are generated downtown and thus more vehicles are deployed. Under dynamic demand, the algorithm has an acceptance rate of new requests of about 90% while a penalty is given to customers who cancel a ride. Analysis showed that customers are rejected if an empty vehicle has to rebalance to their location unless they are premium. In terms of the computational efficiency the algorithm is able to handle between 40-50 requests simultaneously.