A carbon dioxide capture system includes a first heat exchanger that exchanges heat between an exhaust stream and a lean carbon dioxide effluent stream. The carbon dioxide capture system also includes a second heat exchanger in flow communication with the first heat exchanger. The second heat exchanger is configured to cool the exhaust stream such that a condensate is formed, and the second heat exchanger is configured to channel a condensate stream for injection into the lean carbon dioxide effluent stream. A first turboexpander including a first compressor is driven by a first turbine. The first compressor is coupled in flow communication with the first heat exchanger. The first turbine is coupled in flow communication with the first heat exchanger and configured to expand the lean carbon dioxide effluent stream. The carbon dioxide capture system further includes a carbon dioxide membrane unit coupled in flow communication with the first compressor.