Basics of Fluid Mechanics, 2014a

22 CHAPTER 1. INTRODUCTION TO FLUID MECHANICS
1.6.1 Fluid Density
ρ [ ]
kg
m 3
1010
1005
1000
995
990
Water Density As A Function of Temperature and Pressure
} {{ } }
∂P
∂T
dT
∂P
∂ρ
dρ
0.5
1.0
25.0
50.0
75.0
100.0
125.0
150.0
175.0
985
March 15, 2011
0 10 20 30 40 50 60
T[ ◦ C]
Fig. -1.15. Water density as a function of temperature for various pressure. This figure
illustrates the typical situations like the one that appear in Example 1.9
The density is a property that is simple to analyzed and understand. The density
is related to the other state properties such temperature and pressure through the
equation of state or similar. Examples to describe the usage of property are provided.
Example 1.9:
A steel tank filled with water undergoes heating from 10 ◦ Cto50 ◦ C. The initial pressure
can be assumed to atmospheric. Due to the change temperature the tank, (strong steel
structure) undergoes linear expansion of 8 × 10 −6 per ◦ C. Calculate the pressure at the
end of the process. E denotes the Young’s modulus 6 . Assume that the Young modulus
of the water is 2.15 × 10 9 (N/m 2 ) 7 . State your assumptions.
Solution
The expansion of the steel tank will be due to two contributions: one due to the
thermal expansion and one due to the pressure increase in the tank. For this example,
it is assumed that the expansion due to pressure change is negligible. The tank volume
change under the assumptions state here but in the same time the tank walls remain
6 The definition of Young’s modulus is E = σ where in this case σ can be estimated as the pressure
ɛ
change. The definition of ɛ is the ratio length change to to total length ΔL/L.
7 This value is actually of Bulk modulus.