Question 1.P.14: Heat is transferred from condensing steam to a vertical surf...
Heat is transferred from condensing steam to a vertical surface and the resistance to heat transfer is attributable to the thermal resistance of the condensate layer on the surface.
What variables are expected to affect the film thickness at a point?
Obtain the relevant dimensionless groups.
For streamline flow it is found that the film thickness is proportional to the one third power of the volumetric flowrate per unit width. Show that the heat transfer coefficient is expected to be inversely proportional to the one third power of viscosity.
For a film of liquid flowing down a vertical surface, the variables influencing the film thickness δ, include: viscosity of the liquid (water), μ; density of the liquid, ρ; the flow per unit width of surface, Q, and the acceleration due to gravity, g. Thus: \delta=f(\mu, \rho, Q, g).
The dimensions of each variable are: \delta= L , \mu= M / L T , \rho= M / L ^3, Q= L ^2 / T, and g =L/T². Thus, with 5 variables and 3 fundamental dimensions, (5-3)=2 dimensionless groups are expected. Taking \mu, \rho and g as the recurring set, then:
\mu \equiv M / L T , \quad M =\mu L T\rho \equiv M / L ^3, \quad M =\rho L ^3 \therefore \rho L ^3=\mu L T , \quad T =\rho L ^2 / \mu
g \equiv L / T ^2=\mu^2 L / \rho^2 L ^4=\mu^2 / \rho^2 L ^3 \quad \therefore L ^3=\mu^2 / \rho^2 g and L =\mu^{2 / 3} /\left(\rho^{2 / 3} g^{1 / 3}\right)
∴ T =\rho\left(\mu^2 / \rho^2 g\right)^{2 / 3} / \mu=\mu^{1 / 3} /\left(\rho^{1 / 3} g^{2 / 3}\right)
and: M =\mu\left(\mu^2 / \rho^2 g\right)^{1 / 3}\left(\mu^{1 / 3} /\left(\rho^{1 / 3} g^{2 / 3}\right)\right)=\mu^2 /(\rho g)
Thus, dimensionless group 1: Q T / L ^2=Q\left(\mu^{1 / 3} /\left(\rho^{1 / 3} g^{2 / 3}\right)\right) /\left(\mu^{4 / 3} /\left(\rho^{4 / 3} g^{2 / 3}\right)\right)=Q \rho / \mu
dimensionless group 2: \delta L =\delta \mu^{2 / 3} /\left(\rho^{2 / 3} g^{1 / 3}\right) or, cubing =\delta^3 \rho^2 g / \mu^2
and: \underline{\underline{\left(\delta^3 \rho^2 g / \mu^2\right)= f (Q \rho / \mu)}}
This may be written as: \left(\delta^3 \rho^2 g / \mu^2\right)=K(Q \rho / \mu)^n
For streamline flow, \delta \propto Q^{1 / 3} or n = 1
and hence: \left(\delta^3 \rho^2 g / \mu^2\right)=K Q \rho / \mu, \delta^3=K Q \mu /(\rho g) and \delta=(K Q \mu / \rho g)^{1 / 3}
As the resistance to heat transfer is attributable to the thermal resistance of the condensate layer which in turn is a function of the film thickness, then: h \propto k / \deltawhere k is the thermal conductivity of the film and since \delta \propto \mu^{1 / 3}, h \propto k / \mu^{1 / 3}, that is the coefficient is inversely proportional to the one third power of the liquid viscosity.