Mechanical Engineering

Mechanics : Efficiency 

The efficiency of a machine can be described more easily by a mathematical formula rather than by words. It is defined as follows (very simple, elementary school math).

If you translate this formula into plain words, it can be described as follows

Efficiency of a machine is the ratio of the input work (or energy) and output work(energy). It is a kind of indicate that shows how well a machine convert the incoming(or supplied) work (or energy) into another form of work(or energy).

Depending on the type of the information given to you, you can estimate the efficiency using the ratio of the power as shown below.

Why the calculation of efficiency is difficult ?

Before you try to find any answer to this, you would ask me back 'You said.. the efficiency definition is very simple. If it is the case, why you asking about the difficulties of the calculation ?'.

Yes, I remember I said 'the definition is simple', but 'simple definition' does not always mean 'simple calculation'. Depending on what kind of information is given, it would be very simple or very complicated to calculate the efficiency.

If you are given the explciit value for Wout and Win with the same unit, it is just a matter of elemenary school math to calculate the efficiency. However, you would find it difficult to estimate (calculate) the efficient in cases described below.

Case 1 : Wout and Win values are directly given, but the unit is different.

In this case you need to do proper unit conversion, but sometimes the unit conversion would not be that easy.

Case 2 : The form of energy for Wout and Win is different.

For example, if you are trying to calculate the efficiency of an engine that burns gasolin and you are given the information on Win in the form of the amount of gasoline (chemical energy) and Wout in the form of horse power generated by the engine.

Case 3 : The system is made up of multiple components.

In this case, you may go through the complicated system analysis process to get the value of Wout. This is the most difficult cases for efficiency calculation. If you are working as a professional engineer, this would be the case you mostly come across.

Example 1 > The efficiency of a Pump

Let's suppose we have a pump that is flowing the water as shown below. Various information about the pump and water flow are given. Estimate the efficiency of this pump.

Solution >

First write down the efficiency calculation equation and see if you have all the information to evaluate the equation.

You would notice that this calculation is very simple because the input power and output power of the pump is directly given. You don't need to use any other information (One of the most important step in solving any problem is to find which information is necessary and which one is not required for the problem solving. You might not have worried much about this in solving many of the textbook problem because those textbook problem tend to give only those information that are required to solve the problem. However real life problem would not be presented as in textbook problem).

Example 2 > The efficiency of a Pump

Let's suppose we have a pump that is flowing the water as shown below. Various information about the pump and water flow are given. Estimate the efficiency of this pump.

Solution >

This is exactly same system as in previous example, but the calculation of the efficiency would be much complicated than the previous example since the informaion given in the problem is a little bit different.

First write down the efficiency calculation equation and see if you have all the information to evaluate the equation.

As shown below, we don't have enough information that can calculate the efficiency directly.

Since the output power is not directly given, we have to figure this out first. The output power of the pump can be estimated by following equation.

You can figure out each of the parameter values to calculate the output power of the pump as described below.

The problem is how to calculate h_pump. Calculating this parameter is a pretty much complicated. I put the steps to calcuate this parameter in an example of SFEE.

Now you have all the values to calulate the Pout for the pump and figure out the specific number by plugging in all those numbers into the equation as shown below.

Since the input power of the pump is given in the unit of HP (Horse Power), let's convert the P value into HP unit as shown below.

Last step is just elementary school math as shown below.

Reference :

[1]