Thermodynamics – Stirling Engine

1. The problem statement, all variables and given/known data

In 1816, Robert Stirling, a Scottish clergyman, patented
the Stirling engine, which has found a wide variety of applications
ever since. Fuel is burned externally to warm one
of the engine’s two cylinders. A fixed quantity of inert gas
moves cyclically between the cylinders, expanding in the
hot one and contracting in the cold one. Consider
n mol of an ideal monatomic gas being taken once
through the cycle, consisting of two isothermal processes
at temperatures 3Ti and Ti and two constant-volume
processes. Determine in terms of n, R, and Ti (a) the net
energy transferred by heat to the gas and (b) the efficiency
of the engine. A Stirling engine is easier to manufacture
than an internal combustion engine or a turbine.
It can run on burning garbage. It can run on the energy
of sunlight and produce no material exhaust.

2. Relevant equations
w=-∫PdV
e= w/Qh

3. The attempt at a solution
So for part b I’m a little bit confused as to how to calculate work.

Since two of the processes are isovolumetric the work done by them = 0.
Now this is where I get confused. Doesn’t the isothermal process that represents a decrease in the volume translate to work done ON the gas – therefore it should be positive. The solutions manual lists them both with the same sign. If I add my work together I do get the same result (-nRTi ln(4)). So my question I suppose is more conceptual.
How does a decrease in volume represent work done by the engine?

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