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Chapter 10: Problem 82

Saturated water vapor at a pressure of \(12.4 \mathrm{kPa}\) is condensed over 100 horizontal tubes in a rectangular array of 5 tubes high and 20 tubes wide, each with a diameter of \(8 \mathrm{~mm}\). If the tube surfaces are maintained with a uniform temperature of \(30^{\circ} \mathrm{C}\), determine the condensation rate per unit length (in \(\mathrm{kg} / \mathrm{s} \cdot \mathrm{m}\) ) of the tubes.

Short Answer

Expert verified
#tag_title#Answer#tag_content# The condensation rate per unit length (m_dot) of the tubes can be calculated using the provided formula: m_dot = (h * (T_sat - T_surface))/(h_fg * (ρ_steam - ρ_surface)) Plug in the values obtained from the previous step calculations to determine the condensation rate per unit length of the tubes in kg/s*m.

Step by step solution

01

Find steam properties at given pressure

Find the saturated steam properties at 12.4 kPa from steam tables or any reliable source. Note down the saturation temperature (T_sat) and steam density (ρ_steam) at this pressure.
02

Find air properties at the tube surface temperature

Find the film temperature T_film which is the average temperature of the steam on the tube surface and the saturated steam. T_film can be calculated as follows: T_film = (T_sat + T_surface) / 2 Find the air/vapor properties (density, specific heat, thermal conductivity, and dynamic viscosity) at T_film using air property tables or any reliable source.
03

Calculate Non-dimensional Prandtl (Pr) and Schmidt (Sc) numbers

The Prandtl number (Pr) and Schmidt number (Sc) are dimensionless numbers used to determine the mass transfer coefficient. Calculate these numbers as follows: Pr = (Cp * μ) / k Sc = (μ / ρ) / D_AB Where, Cp is specific heat at T_film, μ is dynamic viscosity at T_film, k is thermal conductivity at T_film, ρ is the density of the air/vapor mixture at T_film, D_AB is the mass diffusivity of water vapor in air (you may use a value of \(2.5 \times 10^{-5} \mathrm{~m^2/s}\) for this case).
04

Calculate the heat transfer coefficient (h) using Nusselt number (Nu) relationships

Calculate the heat transfer coefficient (h) from the following Nusselt number (Nu) formula for condensation over horizontal tubes: Nu = 0.725 (Re)*(Pr)^(1/3) Where Re is the Reynolds number calculated as follows: Re = (4 * m_dot) / (π * D * μ) m_dot is the mass flow rate per unit length of the tube. We don't know the value of m_dot yet, so we will come back to this step later after we have a calculated value for it.
05

Calculate mass transfer coefficient (k_m) using Sherwood number (Sh) relationships

Calculate the mass transfer coefficient (k_m) from the following Sherwood number formula for condensation over horizontal tubes: Sh = 0.725 (Re)*(Sc)^(1/3) Where Re is the same Reynolds number from step 4. Using the calculated Sh number and mass diffusivity (D_AB), calculate k_m as follows: k_m = (Sh * D_AB) / D Where D is the diameter of the tube.
06

Calculate total condensation heat transfer (Q) from heat transfer coefficient (h)

Calculate the total heat transfer (Q) using heat transfer coefficient: Q = h * A * (T_sat - T_surface) Where A is the heat transfer area (A = π * D * L). Since we need the condensation rate per unit length of the tube, we can simplify the equation as: Q = h * π * D * (T_sat - T_surface)
07

Calculate total condensation mass transfer (m_dot) from mass transfer coefficient (k_m)

Calculate the total mass transfer (m_dot) by using the mass transfer coefficient: m_dot = k_m * A * (ρ_steam - ρ_surface) Again, since we need the condensation rate per unit length of the tube, we can simplify the equation as: m_dot = k_m * π * D * (ρ_steam - ρ_surface)
08

Equate total heat transfer (Q) with latent heat of condensation times mass transfer (m_dot)

Q = m_dot * h_fg Where h_fg is the latent heat of condensation of the steam at the given pressure. Now, we have the equation: h * π * D * (T_sat - T_surface) = k_m * π * D * (ρ_steam - ρ_surface) * h_fg
09

Calculate m_dot

We have all the required properties except one, the mass flow rate per unit length (m_dot). Rearrange the equation in step 8 and express m_dot in terms of known quantities: m_dot = (h * (T_sat - T_surface))/(h_fg * (ρ_steam - ρ_surface)) Substitute the values of h, T_sat, T_surface, h_fg, ρ_steam, and ρ_surface and calculate the condensation rate per unit length (m_dot) of the tubes in kg/s*m.

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Most popular questions from this chapter

A \(1.5-\mathrm{m}\)-long vertical tube is used for condensing saturated steam at \(60^{\circ} \mathrm{C}\). The surface temperature of the tube is maintained at a uniform temperature of \(40^{\circ} \mathrm{C}\) by flowing coolant inside the tube. Determine the heat transfer rate to the tube and the required tube diameter to condense \(12 \mathrm{~kg} / \mathrm{h}\) of steam during the condensation process. Assume wavy-laminar flow and that the tube diameter is large relative to the thickness of the liquid film at the bottom of the tube. Are these good assumptions?

Saturated steam at \(100^{\circ} \mathrm{C}\) condenses on a 2-m \(\times 2-\mathrm{m}\) plate that is tilted \(40^{\circ}\) from the vertical. The plate is maintained at \(80^{\circ} \mathrm{C}\) by cooling it from the other side. Determine (a) the average heat transfer coefficient over the entire plate and \((b)\) the rate at which the condensate drips off the plate at the bottom. Assume wavy-laminar flow. Is this a good assumption?

Discuss some methods of enhancing pool boiling heat transfer permanently.

In condensate flow, how is the wetted perimeter defined? How does wetted perimeter differ from ordinary perimeter?

Saturated water vapor at \(40^{\circ} \mathrm{C}\) is to be condensed as it flows through a tube at a rate of \(0.2 \mathrm{~kg} / \mathrm{s}\). The condensate leaves the tube as a saturated liquid at \(40^{\circ} \mathrm{C}\). The rate of heat transfer from the tube is (a) \(34 \mathrm{~kJ} / \mathrm{s}\) (b) \(268 \mathrm{~kJ} / \mathrm{s}\) (c) \(453 \mathrm{~kJ} / \mathrm{s}\) (d) \(481 \mathrm{~kJ} / \mathrm{s}\) (e) \(515 \mathrm{~kJ} / \mathrm{s}\)
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