Willian's law (throttle-governed steam engine) — dependence of hourly steam consumption According to Willian's law, for a throttle-governed engine, the steam consumption per hour is proportional to which function of indicated power (I.P.)?

Introduction / Context:
Willian's law is an empirical relationship used to assess steam rate and efficiency trends of throttle-governed steam engines. It allows linear extrapolation of steam consumption against indicated power to estimate no-load steam usage and frictional losses.

Given Data / Assumptions:

• Throttle (not cut-off) governing — steam pressure at the cylinder is throttled as load changes.
• Hourly steam consumption m and indicated power I.P. are measured over several loads.
• Linearity holds within the tested operating range.

Concept / Approach:
The Willian's line is typically expressed as m = a + b * I.P. where a is the intercept representing no-load steam consumption (to overcome mechanical and pumping losses) and b is the slope representing steam rate per unit indicated power. Thus, m varies linearly with I.P., i.e., is proportional to I.P. once the constant offset a is recognized.

Step-by-Step Solution:

Verification / Alternative check:
Plotting m versus I.P. yields a straight line — the Willian's line. Extrapolation to I.P. = 0 gives intercept a (no-load steam), while slope b provides specific steam consumption per unit indicated power.

Why Other Options Are Wrong:

• sqrt(I.P.) or (I.P.)^2 or 1/I.P.: not supported by throttle-governed empirical data.
• Constant-only forms ignore the clear load dependence captured by b * I.P.

Common Pitfalls:
Applying Willian's law to cut-off governed engines (where relation is not linear); misinterpreting the intercept as zero; mixing indicated with brake power when plotting steam rate.

Final Answer: