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The molar flow rate of a chemical species is a measure of how many moles of that substance pass by a given point per unit time. It is a key parameter in process engineering and understanding the scale of a chemical reaction or process. When we talk about a substance flowing at, say, 35 kmol/h, that's the molar flow rate.

For a mixture, the individual molar flow rates of each component need to be determined to adequately describe the system. For instance, if you have a stream with multiple components like hydrogen and 1-butene, you calculate the molar flow rate of each based on their mole percentage in the stream. Once you have these values, they can be used to do further calculations necessary for process design such as sizing equipment or pipes which is integral for proper operation and safety.

The Ideal Gas Law is a fundamental equation in thermochemistry and fluid dynamics that relates the pressure, volume, temperature, and number of moles of an idealized gas. It's represented as PV = nRT, where P is the pressure of the gas, V is the volume it occupies, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.

When applied to real-world situations, the Ideal Gas Law allows for approximation in determining volumes or pressures. This is especially useful when we are determining pipe diameters, as in the example problem, since the volume flow rate can be calculated using this law if the gas behaves close to ideal conditions—which often applies to gases at low pressure and high temperature.

Fluid dynamics is governed by conservation laws, and the continuity equation is the mathematical expression that embodies the conservation of mass in a fluid flow. It states that the mass flow rate must remain constant from one cross-section of a pipe to another. In simple terms, it means what goes in must come out, assuming steady-state flow without any accumulation happening within the pipe.

When using the continuity equation for gases, it's often expressed in terms of volume flow rate and velocity. For instance, if you have a certain volume flow rate of a gas and a corresponding speed at which it moves through a pipe, the equation helps to compute the pipe's cross-sectional area, which can then be used to calculate the pipe diameter. This aspect is crucial for designing piping systems in chemical processes where size and flow rates affect the performance and efficiency.

In the realm of gas mixtures, where various gases are blended, Kay's rule provides a practical approach to find an 'equivalent' molar mass for the mixture. It's based on a weighted average where each gas' contribution to the mixture's overall molar mass is proportional to its molar flow rate.

By taking a molar flow rate weighted sum of the individual gas molar masses, you get a representation of the mixture's molar mass, which is then utilized to perform calculations as if the mixture were a single gas. This simplification is key in the application of the Ideal Gas Law to mixtures, to predict the behavior of the mixed gas under various conditions—like in our example problem to help calculate the appropriate pipe diameter required for transporting the gas mixture.