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Dimensional Analysis/Model Testing
You are tasked with designing a heat exchanger around a section of piping in a synthesis plant in which temperature control will be critical to prevent bi-product formation.
In this process the temperature difference across the given length of pipe can be described as:
ΔT= ΔT(ρ, μ, V, C, Q, D)
where density, ρ[kg/m^3], viscosity, μ[kg/(ms)], velocity V[m/s], and heat capacity C [J/(kgK)] are fluid properties in the real system. D [m] is the diameter of the pipe, and Q [J/s] is the total rate of heat input from the heat exchanger along entire surface area of the pipe considered. [reminder: J=kg*m^2/s^2]
a) How many dimensionless groups are there? 7 variables – 4 dimensions = 3 dimensionless groups N1, N2, N b) Use Buckingham Pi theorem to find all the dimensionless groups in terms of ρ, V, C, D?
Variable Units M L T K ρ M/L^3 1 -3 0 0 μ M/(L*T) 1 -1 -1 0 V L/T 0 1 -1 0 C L^2 /(T^2 K) 0 2 -2 - Q ML^2 /T^3 1 2 -3 0 D L 0 1 0 0 ΔT K 0 0 0 -
N1= μρaVbDcCd=M^0 L^0 T^0 K^0 K: d= M: 1+a=0 a=- L: -1-3a+b+c=0 c=- T: -1-b=0 b=- N1=μ/(ρVD) or (ρVD)/μ
N2= QρaVbDcCd=M^0 L^0 T^0 K^0 K: d= M: 1+a=0 a=- T: -3-b=0b=- L: 2-3a+b+c=0 c=- N2= Q/(ρD^2 V^3 )
N3= ΔT(ρaVbDcCd)=M^0 L^0 T^0 K^0 K: 1-d=0 d= M: a= T: -b-2d=0b=- L: -3a+b+c+2d=0 b+c=-2 c= N3= (CΔT)/V^2
c) You would like to run a model test on a cheaper fluid. The D and ΔT are kept the same in the model system as the real setup. There is a selection of model fluids to choose from that all have Cmodel =0.25Creal and ρmodel =0.5ρreal. Find the required model fluid viscosity, μmodel and velocity, Vmodel, along with the needed heat input from the exchanger, Qmodel, in terms of μreal, Vreal, Qreal to satisfy similarity conditions.
N1, N2, N3 must be same between model and real system
First find required V from N3: Cmodel =0.25Creal and ρmodel =0.5ρreal, ΔTreal=^ ΔTmodel,Dreal= Dmodel (Creal ΔTreal)/Vreal^2 =(Cmodel ΔT (^) model)/Vmodel^2 (Vmodel/Vreal)^2 =Cmodel/Creal Vmodel=0.5Vreal
To find Q, use N2: QrealDreal^8 /(ρrealVreal^3 )= QmodelDmodel^8 /(ρmodelVmodel^3 ) Dreal= Dmodel, and ρmodel =0.5ρreal, Vmodel=0.5Vreal
Qreal/Qmodel= (ρrealVreal^3 )/(ρmodelVmodel^3 ) Qreal/Qmodel=
To find μ for model system use N1 to satisfy similarity condition: (ρrealVrealDreal)/μreal=^ (ρmodelVmodelDmodel)/μmodel Dreal= Dmodel, and ρmodel =0.5ρreal, Vmodel=0.5Vreal μmodel=0.25μreal
e. If instead of an infinite fluid reservoir, the sphere is in the center of a vertical coffee stirrer straw (cylindrical tube) of diameter 3mm, calculate the terminal velocity of the particle. You may use Ø = 1 + 2.10. (5 points)
Ø = 1 + 2.10* (0.1 mm / 3mm ) = 1. CD = 24/Re * Ø, therefore Vp = 1/18 * (ρ- ρp) * Dp^2 |g|/ η * 1/ Ø = 0.00544/1.07 = 0.00508 m/s
f. A second CO 2 bubble rises through the liquid. Treat the liquid as infinite. If instead of the bubble diameter we know the bubble velocity, show how we can use the CD vs. Re plot below to determine the bubble diameter. Show work to justify your method. (5 points)
P 1 = 3.45 x 10^5 Pa D 1 = D 2 = 2 cm P 2 = 2.07 x 10^7 Pa D 3 = 1.07 mm Q = 1.24 x 10-4^ m^3 /s
a) What is the power requirement for the pump in this system? (17 Points)
Mass Balance: ρ<v 2 > 1 A 1 = ρ<v 2 >A 2 => <v 1 >=<v 2 >
Engineering Bernoulli Equation: ∫
Calculating Power: ( ) ( )
ϴ
ϴ
(^1 )
3
4
x
y
Depiction of Jet of Water 3
4
