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Two-Phase Flow and Heat Transfer

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Fundamentals of Multiphase Heat Transfer and Flow

Abstract

This chapter starts with definitions of various parameters for two-phase flow and flow patterns in vertical and horizontal tubes. This is followed by two-phase flow models as well as prediction of pressure drops and void fractions. Finally, the two-phase flow regimes and heat transfer characteristics for forced convective condensation and boiling at both macro- and microscale levels are presented.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Connecticut, Storrs, CT, USA

    Amir Faghri

  2. Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO, USA

    Yuwen Zhang

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  1. Amir Faghri
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  2. Yuwen Zhang
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Corresponding author

Correspondence to Amir Faghri .

Problems

Problems

  1. 10.1.

    Express slip ratio in terms of the total cross-sectional area, total volume flow rate, void fraction, and vapor velocity.

  2. 10.2.

    Redo Problem 10.1 using the mass flux instead of the total volume flow rate.

  3. 10.3.

    Express volumetric flow fraction in terms of the total cross-sectional area, total volume flow rate, and gas superficial velocity.

  4. 10.4.

    Consider liquid–vapor two-phase flow. Find the expression for mass flux as a function of the quality, the individual phase velocities, and the densities.

  5. 10.5.

    Express total volumetric flux in terms of the individual mass flow rates, the total cross-sectional area, and the phase densities.

  6. 10.6.

    Determine the flow regime for upward flow of 10 kg/s of R-11 in a vertical 0.1 m diameter tube at 10 bar and 25% quality.

  7. 10.7.

    For a horizontal miniature tube of 1 mm inner diameter, determine the flow regime for 1.5 × 10−3 kg/s of water–steam at 1 atm and 10% quality.

  8. 10.8.

    For a horizontal circular tube with inner diameter of 5 cm (D = 0.05 m), determine the flow regime for 1.5 × 10−3 kg/s of steam‐water flow at 1 atm and 59% quality.

  9. 10.9.

    The flow of 0.1 kg/s of air and 5 kg/s of water at 20 °C and 1 atm at a given point in a horizontal channel is in the bubbly regime. Describe qualitatively how the two-phase flow should change to achieve stratified flow.

  10. 10.10.

    Determine the pressure drop due to friction for a 1 kg/s flow of steam–water in a 10 cm inner diameter tube at 2 atm and 40% quality.

  11. 10.11.

    Calculate the void fraction using the homogeneous model for a two-phase water–steam flow with a quality of 35% at 15 bars.

  12. 10.12.

    Specify the governing equations for adiabatic two-phase flow in a circular tube of constant cross-sectional area using (1) the homogeneous model and (2) the separated flow model.

  13. 10.13.

    A capillary tube-suction line heat exchanger consists of a capillary tube and a suction line welded together as shown in Fig. P10.13. While evaporation takes place in the capillary tube, the suction line flow is single phase. Specify the governing equations for fluid flow and heat transfer in the system using the homogeneous model.

    figure g

    Fig. P10.13

  14. 10.14.

    A mixture of steam–water flows in a 5-cm inner diameter vertical tube at 10 kg/s, 5 atm, and 10% quality. What is the void fraction?

  15. 10.15.

    A mixture of steam–water flows in a 5-cm inner diameter vertical tube at 1 atm. The superficial velocities of the liquid and vapor are 1 and 100 m/s, respectively. Determine the void fraction using Eq. (10.105).

  16. 10.16.

    Steam condenses at 80 °C as it flows inside a horizontal tube with an inner diameter of 5 cm. The mass flow rate is 0.5 kg/s, and the quality is x = 0.75. Find the heat transfer coefficient.

  17. 10.17.

    Calculate the flow boiling two-phase heat transfer coefficient for a 400 kg/sm2 of R22 at a saturation temperature of −20 °C and 20% quality in an upward vertical tube with an inner diameter of 1 cm and a wall temperature of 0 °C.

  18. 10.18.

    Saturated liquid water at 1 atm enters a vertical tube evaporator (circular cross section) with a constant heat flux of \(q^{\prime\prime }\) (see Fig. P10.18). The tube inner diameter is D, and the thickness of the tube can be neglected (i.e., assume that the outer diameter of the tube is equal to the inner diameter).

    figure h

    Fig. P10.18

    1. a.

      Determine the quality of mixture as a function of distance from the inlet if the kinetic and potential energy can be neglected.

    2. b.

      Assuming homogeneous flow model is valid, determine the void fraction as a function of distance from the inlet.

    3. c.

      Develop an expression for (but not evaluate) the gravitational pressure drop across the tube for a given length, L.

    4. d.

      If the mass flow rate is 0.05 kg/s, heat flux is 100 kW/m2, and the inner diameter of the tube is 2 cm, determine the minimum length of the tube to completely evaporate the liquid.

  19. 10.19.

    A stainless steel heat exchanger (Fig. P10.19) is being used as a flow boiler. The evaporant flows through 10 tubes with a liquid counterflow. Find the heat transfer coefficient when the quality, x, is 0.5. The evaporant is ammonia at 200 K, and the evaporant passage effective diameter is 1 mm. The wall heat flux is 5000 W/m2.

    figure i

    Fig. P10.19

  20. 10.20.

    You are given a flow boiler that is fed with saturated liquid ammonia at 200 K. 100% saturated vapor exits the boiler. The boiler has 547 passages, each of which is 1.25 mm high and 1.25 mm wide. The mass flow rate of the ammonia is 11.45 kg/hr (see Fig. P10.20). Determine the phase velocity at the beginning and the end of the flow boiler. What is the percentage increase in phase velocity from the beginning to the end of the boiler?

    figure j

    Fig. P10.20

  21. 10.21.

    In an electronic cooling system, a row of parallel tubes with diameters of D are embedded in a horizontal plate with thickness W (Fig. P10.21). A mixture of liquid and vapor with void fraction of \(\alpha\) flows in the tube and the flow pattern is stratified. The upper surface of the plate is heated at constant heat flux while the bottom of the plate is insulated. The temperature of both liquid and vapor inside the tube is Tsat, but the heat transfer coefficients for liquid and vapor are different. Since the structure is symmetric, it is sufficient to consider only one cell with a width of S. Specify the governing equations and corresponding boundary conditions for heat conduction in the plate. You can use either the Cartesian or the cylindrical coordinate system.

    figure k

    Fig. P10.21

  22. 10.22.

    The spacing between the tubes in Problem 10.21 is a critical parameter for the performance of the electronic cooling system. While small spacing allows more tubes per unit width, it also increases the conductance resistance from the top of the plate to the bottom of the plate. An optimal spacing will minimize the maximum temperature in the system. Write a computer program to solve the above conduction problem for the following parameters: D = 1 mm, W = 2 mm, \(\alpha = 0.5,\) \(q^{\prime\prime} = 10^{6} \,{\text{W/m}}^{2}\), \(h_{\ell } = 10^{4} \,{\text{W}}/{\text{m}}^{2} {\text{K}}\), \(h_{\text{v}} = 100\,{\text{W/m}}^{2} {\text{-K}}\). Perform a parametric study to find the optimal spacing for the system in Problem 10.21.

  23. 10.23.

    A pulsating heat pipe (PHP) is a long miniature tube bent into many turns with the evaporator and condenser sections located at these turns (see Sect. 1.5.4). Since the diameter of the PHP is very small (less than 5 mm), the working fluid charged in the PHP forms vapor plugs and liquid slugs due to capillary action. Specify the governing equation to describe the oscillatory motion of an arbitrary liquid slug with a length of L in the PHP. The effects of surface tension and friction must be taken into account.

  24. 10.24.

    Evaporation and condensation on the thin film left behind by the liquid slug are the mechanisms that sustain the pressure difference between the two ends of the liquid slug. If the vapor phase satisfies the ideal gas law, obtain the energy equation for the vapor plug. Note that the mass of the vapor plug is not constant due to evaporation and condensation.

  25. 10.25.

    For a miniature passage with a very large length to width ratio, a fully developed flow in both vapor and liquid is very typical. A cross section of a typical miniature passage with axial rectangular grooves is shown in Fig. P10.25a. The liquid flows in the capillary microgrooves and the vapor flows in the central part of the passage free of liquid. Fluid flow in the grooved passage can be analyzed by considering a computational domain as shown in Fig. P10.25b. The vapor and liquid flow along the z-coordinate (concurrently or countercurrently). It can be assumed that (1) the fluid flow is laminar with fully developed velocity profiles, (2) the curvature of the liquid–vapor interface is constant, and (3) symmetry conditions are applicable at the three exterior boundaries of the vapor domain. Develop a steady-state mathematical model for the liquid and vapor flows that includes coupled vapor and liquid flows with shear stresses at the liquid free surface due to the vapor–liquid frictional interaction.

    figure l

    Fig. P10.25

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Faghri, A., Zhang, Y. (2020). Two-Phase Flow and Heat Transfer. In: Fundamentals of Multiphase Heat Transfer and Flow. Springer, Cham. https://doi.org/10.1007/978-3-030-22137-9_10

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