Density for liquid and vapor phases equation
Rho = (-7.981*10^-19)*(T^9) + (7.002*10^-16)*(T^8) – (2.087*10^-13)*(T^7) + (1.821*10^-11)*(T^6) + (1.971*10^-9)*(T^5) – (3.474*10^-7)*(T^4) – (3.29*10^-6)*(T^3) + 0.001316 * (T^2) - 0.9326 * T + 884.5
I was searching the internet for the mathematical model for a pump, and I found this equation in the attached image. This equation is for finding the density in order to find pump work. My question is, can someone please explain these numbers or from where this equation was derived? Or at least help me find a reference for this equation because I found it on the Internet without a source.
Thank you and I appreciate your help
This is a curve fit for the density of some liquid. It doesn't appear to have the right shape for water (see figure and attached spreadsheet). This is only part of calculating pump work, which includes enthalpy and entropy, which you will need tables or functions for the liquid in question. You first calculate the isentropic work by subtracting the enthalpy at the new pressure and same entropy from the original state. You then divide by the isentropic efficiency to arrive at the actual work per unit mass. The actual compressed state is the new pressure at the new enthalpy, which gives you the new temperature. I refer you to any textbook on thermodynamics for the discussion.
I have lots of properties and spreadsheets for fluids and organic Rankine cycles as well as years of design and testing experience.
The equation is a mathematical model that relates temperature to density. It appears to be a polynomial equation with terms of different powers of T. Each term is multiplied by a coefficient (a numerical value). The equation might have been derived through empirical fitting or regression analysis, where experimental data or other information was used to determine the coefficients. It's difficult to determine the exact origin or source of the equation without further information.
I have attached an organic Rankine cycle using R114 and a spreadsheet with 39 different refrigerants. This book is free today (Friday 5/19): https://read.amazon.com/kp/embed?asin=B084HKJTDZ&preview=newtab&linkCode=kpe&ref_=cm_sw_r_kb_dp_8HK3Y6D0R45KFC1TRJSF
Here's something you can do with the spreadsheet property functions: compare the above density curve to 29 possible refrigerants. Result: none of them fits so it must be something else. The closest one is R718.
Correction
Rho = (-7.981*10^-19)*(T^9) + (7.002*10^-16)*(T^8) – (2.087*10^-13)*(T^7) + (1.821*10^-11)*(T^6) + (1.971*10^-9)*(T^5) – (3.474*10^-7)*(T^4) – (3.29*10^-6)*(T^3) + 0.001316 * (T^2) - 0.9326 * T + 884.5
I found out this equation is the thermophysical properties for Toluene working fluid for ORC. NOT FOR WATER
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