Balancing units in size-energy relationship

[kppnut]

Homework Statement

Can someone help me balance the units in these equations? They are statistical best fits for product size - energy relationships.

Homework Equations

1. P80 = Cl + W*Sl
With units substituted into equation: micron = micron + kWh/t * Sl

2. P80 = Cp * W^Sp
With units substituted into equation: micron = micron * kWh/t^Sp

The Attempt at a Solution

Sl = micron*t/kWh (correct me if I'm wrong)

Sp is where I get properly lost. I have assumed Cl and Cp are in micron as they are the constants in the best fit statistical relationship representing the intersect with the y-axis (microns) where x = 0 (kWh/t = 0).

Can anyone help me balance the units in this equation?

 

[KataKoniK]

it is important to make sure that the units in an equation are balanced and consistent. This ensures that the equation is mathematically correct and makes sense in the context of the problem. In the first equation, P80 represents a particle size and has units of microns. Cl and Sl represent constants and have units of microns as well. W represents energy and has units of kWh/t (kilowatt-hours per ton). By substituting these units into the equation, we can see that both sides have the same units of microns, which means the units are balanced.

In the second equation, P80 is again the particle size with units of microns. Cp is a constant with units of microns, and W represents energy with units of kWh/t. However, we also have the exponent Sp, which is a dimensionless quantity. This means that the units on the right side of the equation will be microns multiplied by a dimensionless quantity, which is still just microns. Therefore, the units are balanced in this equation as well.

In summary, both equations have balanced units and make sense in the context of the problem. It is important to always check and balance the units in equations to ensure their accuracy and validity.