Transport Phenomena in Multiphase Systems

Amir Faghri and Yuwen Zhang

Chapter 5 SOLID-LIQUID-VAPOR PHENOMENA AND INTERFACIAL HEAT AND MASS TRANSFER - all with Video Answers

Educators

Chapter Questions

Problem 1

Liquid droplets that condense to form fog in the atmosphere can be as small as $2 \mu \mathrm{~m}$ in diameter when they first form. Determine the pressure in a droplet of this size at $20^{\circ} \mathrm{C}(101.3 \mathrm{kPa})$. Note that for water in contact with air, the surface tension $\sigma$ is $0.0728 \mathrm{~N} / \mathrm{m}$. Describe any assumption you need to make in order to get the final answer.

Elan Stopnitzky

Numerade Educator

Problem 1

The temperature at the meniscas of Problem 5.14 equals the saturation temperature, and the bulk liquid is at a lower temperature $T_L$. Repeat Problem

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Problem 2

A 0.2 -mm-thick water film sits on a surface held at a temperature of $60^{\circ} \mathrm{C}$. The liquid film is exposed to air at a bulk temperature of $T_6=20^{\circ} \mathrm{C}$, and the convective heat transfer coefficient between the liquid film and the air is $h_\theta$ $=15 \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}$. Suppose the wave number is $\dot{\alpha}=2$. What is the critical Marangoni number, $M a_e$ ? Is the liquid film stable?

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03:19

Problem 3

A 0.2 -mm-diameter tube is vertically placed in a pool of water at $20{ }^{\circ} \mathrm{C}$. Find the capillary rise in tubes of the following materials: (a) aluminum, (b) brass, (c) copper, and (d) steel.

Eric Mockensturm

Numerade Educator

02:07

Problem 4

Two nearly vertical but nonparallel plates touch a pool of water with their parallel bottom edges. The angle between the plates is $2 \gamma 50$ that these plates would intersect somewhere under the pool surface. The distance between the plates at the pool surface level, $2 W$, is small. The apparent contact angle is denoted by $\theta$. The liquid-vapor meniscus between the plates is elevated from the pool surface level due to capillary pressure. Assume that the meniscus curvature does not change along the liquid-vapor interface. Derive an algebraic equation for the height of the capillary rise $H$ for this situation (for $\theta \neq 0$ and $\gamma \neq 0$ ). Using the derived equation, estimate $H$ for the following data: $W=0.5 \mathrm{~mm}, \theta=0, \gamma=0$, $\rho_{\ell}=1000 \mathrm{~kg} / \mathrm{m}^3, \sigma=0.06 \mathrm{~N} / \mathrm{m}$.

Kudakwashe Mapiki

Numerade Educator

04:43

Problem 5

A small amount of liquid metal resides between two high melting point powder particles separated by a distance of $\ell\left(\ell>2 r_0\right)$ (see Fig. P5.1). If the effect of gravity on the liquid metal is negligible, estimate the force required to hold the two particles together.

Vishal Gupta

Numerade Educator

Problem 6

Write the continuity, momentum, energy, and species equations and the necessary boundary conditions for a very long capillary tube that is open at both ends, as shown in Fig P5.2. The evaporation is driven by the concentration gradient of vapor in the air. The evaporation cools the interface while the wall heats the fluid, causing a temperature gradient along the interface. The assumptions are the same as Example 5.4.

Figure P5. 2

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Problem 7

Consider an ultrathin evaporating liquid film on a bot solid surface of temperature $T_*$ in the presence of a saturated vapor of temperature $T_r$. The film thickness $\delta$ is small and the disjoining pressure $p_d$ is significant. The film is flat. Does the disjoining pressure increase or decrease the evaporation rate in comparison to the hypothetical case where $p_d=0$ (for the same film thickness and temperatures)? Assuming that $T_*-T_v$ is extremely small, explain the effect of $p_d$ on the evaporation rate by referring to the extended Kelvin equation and interfacial resistance.

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Problem 8

A liquid film of ammonia with thickness $\delta=0.375 \times 10^{-3} \mathrm{~m}$ is evaporating from a smooth solid surface into bulk vapor. The solid surface temperature is $T_*=250,12 \mathrm{~K}$. The bulk vapor temperature is $T_r=250 \mathrm{~K}$. The constants in the correlation for disjoining pressure $p_j=-A^{\prime} \delta^{-\delta}$ are $A^{\prime}=10^{-21}$ and $B$ $=3$. The accommodation cocfficient is equal to unity. In this particular case, the perfect gas law can be used instead of the saturation tables, for convenience. Estimate the heat flux at the liquid interface.

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04:38

Problem 9

A slug of water 4.5 mm in length sits in a vertical copper tube with an inner radius of 2 mm at room temperature. The temperature of the whole system is slowly increasing. At what temperature will the water slug start to move?

Yaqub Khan

Numerade Educator

02:14

Problem 10

Derive eq. (5.159) from the momentum balance at the liquid interface.

Amit Srivastava

Numerade Educator

Problem 11

Starting from the appropriate equations of Chapter 3 , derive eq. ( 5.160 ). Show the required control volumes.

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Problem 12

Nusselt-type condensation takes place on a flat vertical plate made from aluminum. The thickness of the plate is 5 mm and there are 1 mm ID micro channels inside it containing cooling liquid. You must increase the rate of condensation, which is restricted primarily by the conduction through the condensate film. What would you do?

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01:28

Problem 13

The effective heat transfer during evaporation of liquid is more extensive from a solid surface with small grooves than from a plain surface. How would you explain this?

Manik Pulyani

Numerade Educator

Problem 14

A vertical capillary tube is connected to bulk liquid at the bottom end and to a heat source at the upper end. The liquid evaporates from the liquidvapor meniscus, so a quasi-steady state exists. What is the heat load that would initiate dry-out of the upper end and lead to unbounded increase of its temperature? Write corresponding general formulas or equations.

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01:05

Problem 16

A capillary tube of inner radius $r$ is vertically dipped into a liquid. The depth from the free surface to the lower end of the tube is $a$. When the liquid rises in the capillary tube due to capillary force, it can be viewed as a variable mass system with capillary force, gravity, and viscous force acting upon it. What is the governing equation for the liquid slug in the capillary tube? The shape of the meniscus in the tube can be assumed to be the same at all times.

Averell Hause

Carnegie Mellon University

Problem 17

Heat transfer in the thin-film region of an axially-grooved evaporator was analyzed in Section 5.5.4. For the condenser region, the effect of interfacial resistance becomes negligible so that the temperature of the liquid vapor interface is equal to the vapor temperature ( $T_b=T_k$ ). It is further assumed that (1) the surface of the liquid film is smooth and the film thickness variation along the s-coordinate is weak (see Fig. 5.15), i.c., $(d \delta / d s)^2 \& 1$, and (2) the disjoining pressure gradient along the film can be neglected in comparison to that of the capillary pressure because of the large film thickness. Analyze the heat transfer on the condenser region based on the above assumptions.

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Problem 18

A typical horizontally oriented rotating miniature heat pipe is shown in Fig. P5.3(a). The fluid flow in the rotating miniature heat pipe is characterized by axial fluid flows and liquid film flows transverse to the axial direction. Axial fluid flows include an axial liquid flow in the groove and a countercurrent vapor flow with varied mass rate due to evaporation and condensation. Liquid film flows transverse to the axial direction occur both in the evaporator and in the condenser. In the evaporatot, an evaporating thin liquid film flowing up the groove side wall is driven by the disjoining pressure and the surface tension. In the condenser, a coadensate film flowing into the groove is sustained by centrifugal force. A change in radius of curvature of the liquid-vapor interface causes a capillary pressure difference between the condenser and evaporator, which promotes the flow of condensate back to the evaporator. In addition, a varying liquid depth allows the centrifugal acceleration to produce a hydrostatic pressure change along the groove that pumps the condensate back to the evaporator. In order to analyze heat transfer in the micro region, the radius of curvature of the liquid-vapor interface in the meniscus region, as shown in Fig. P5.3 (b) is assumed to be constant. The coordinate system for the evaporating thin film flow is shown in Fig. P53(c). The assumptions made are as follows: The vapor temperature, $T_n$ and the wall temperature, $T_n$ are constant; the radius of curvature of the liquid-vapor interface in the meniscus region and the rotational radius of the micro region, $r_{\text {rate }}$ are constant, and the influences of the Coriolis force, gravitational force, and vapor drag on the evaporating thin film flow are negligible. Analyze heat transfer in the evaporator of the rotating miniature heat pipe.

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Problem 19

The cross-sectional view and the coordinate system of the condenser section of a rotating miniature beat pipe are shown in Fig. P5.4. It is assumed that the vapor and liquid flows are incompressible and that the vapor and liquid are in a saturated state. The contact angle of the meniscus, $\theta$, is constant. The effect of gravitational foree on the flow is negligible, and the influences of the Coriolis force and surface tension on the circumferential condensate film flow are negligibly small. Thus, the circumferential film flow is symmetrical about the rotational coordinate of $y$. Assuming that the condensate film thickness is very small compared to $s_{\text {b }}$ the momentum equations in the $y$ and $x$ directions for the condensate film flow and boundary conditions are similar to those used for rotating heat pipes with a large diameter. Derive the ordinary differential equation that governs the liquid film thickness.

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Problem 20

A Newtonian liquid is in an open cylindrical container with a radius of $R$ and beight of $H$ (see Fig. P5.5). A stationary continuous axisymmetric laser beam with an intensity of $I=I_0 \exp \left(-r^2 / r_0^2\right)$ irradiates the top of the liquid surface. The side of the cylindrical container is maintained at a constant temperature, $T_0$, and the bottom of the container is adiabatic. Supposing the surface tension is a linear function of temperature $\sigma=C_0-C_1 T$, write the governing equations and the corresponding boundary conditions of the problem. The effeet of gravity is negligible, and the liquid flow induced by the laser beam can be assumed to be axisymmetric.

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Problem 21

Introduce appropriate nondimensional variables and nondimensionalize the governing equations and boundary conditions in Problem 5.20. Identify the dominant dimensionless variables in the problem.

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Problem 22

Saturated vapor enters a miniature channel with a radius of $R$ and a length of $L$ (see Fig. P5.6). Condensation takes place on the wall of the channel, since the wall temperature $\mathrm{T}_{\mathrm{w}}$ is below the saturation temperature of the working fluid $T_{\text {ss- }}$. The condensate fluid flows in the positive $x$-direction due to the effects of shear force and surface tension. The problem is condensation on the inner surface of the miniature channel with cocurrent vapor flow. The average vapor velocity is not constant along the $x$ direction because condensation occurring on the wall reduces the amount of vapor flow in the core of the tube. Capillary blocking occurs when capillary force causes the liquid to block the tube cross-section at some distance, $L_d$, from the condenser entrance. Supposing the effect of gravity on the two-phase flow in the miniature tube is negligible, write the governing equations and boundary conditions for the liquid and vapor flows.

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Problem 23

Heat is applied to the outer surface of a microchannel with inner and outer radii of $r_i$ and $r_a$, respectively, as shown in Fig. PS.7. Specify the governing equations that describe evaporation in the microchannel. The effects of disjoining pressure and interfacial thermal resistance must be taken into account.

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Problem 24

Evaporation in the evaporator section of a pulsating beat pipe (PHP) with an open end is analyzed in Example 5.6. The physical model of the condenser section of the PHP with an open end is shown in Fig. P5.8. It is assumed that the liquid film in the condenser section is divided into two regions: a thin film region and a meniscus region. These two regions are separated by a transition point where the film thickness is $\delta=\delta_v$. The eurvature of the liquid film in the meniscus region is assumed to be constant. Assuming the effect of disjoining pressure is negligible, specify the governing equation for the liquid film thickness in $0<x<L_2$.

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Problem 25

The fluid mechanics of capillary tubes and microchannels is of great interest for modeling transport phenomena in micro devices. Develop the physical formulation, including continuity and momentum equations with appropriate boundary and initial conditions, for a typical miniature channel with a free surface as depicted in Fig. P5.9 for both submerged and blocked end configuration. Neglect mass transfer and Marangoni effects.

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03:31

Problem 26

Describe the physical/mathematical formulation to model the problem related to instability growth of a free-flowing cylindrical column of fluid as shown in Fig. P5.10.

Surendra Kumar

Numerade Educator

02:25

Problem 27

Obtain an analytical solution for the shape of the meniscus in a capillary tube in 1 g environment for mechanical equilibrium, as shown in Fig, P5.11 for both parallel plates and circular tubes. Assume the effeet of the gaseous region is negligible with vapor pressure constant. You should get a second order ordinary differential equation for $f(y)$ which can be solved by the Runge-Kutta method. Obtain numerical results for the working fluid as acetone with $y_0=0.001 \mathrm{~m}$ and contact angle $\theta=0.0$. Obtain a closed analytical solution for the situation with no gravity.

Amit Srivastava

Numerade Educator

03:31

Problem 28

Obtain an analytical solution for the rise of a meniscus in a capillary tube with both ends open, as shown in Fig. P5.12, for both parallel plates and a circular tube. If $\bar{u}$ is the bulk motion or the average velocity, obtain your solution in terms of $\bar{u}=d \vec{H} / d \overrightarrow{ }$ where $H$ is the mean height of the meniscus.

Surendra Kumar

Numerade Educator

06:53

Problem 29

Consider the steady, laminar flow of a liquid film down an inclined flat plate (see Fig. P5.13). Assume the flow is fully developed. However, the presence of a small temperature gradient in the axial direction $d T / d z$ produces a constant surface tension gradient $d \sigma / d z=C$. Assume the surface tension effect is only at the interface, which produces a nonzero shear stress and no other effect on physical properties of the system. Solve the momentum equation to determine the shear stress variation and velocity distribution as a function of distance perpendicular to the plate.

Khoobchandra Agrawal

Numerade Educator

Problem 30

Consider the absorption of gas $A\left(\mathrm{O}_2\right)$ to a laminar, steady, fully developed, isothermal falling film of liquid $B$ (water) (see Fig. 5.14). Since $\mathrm{O}_2$ is slightly soluble in water, one can assume the properties of water are not affected. Since diffusion of $\mathrm{O}_2$ is taking place slowly and thermal diffusion is in the immediate vicinity of liquid, $\mathrm{O}_2$ does not penetrate very far into the liquid. Obtain the total mole flow rate of gas $A$ to liquid $B$,

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Problem 31

Consider the solid dissolution of a component $A$ for a solid wall to a steady, laminar, isothermal falling liquid film B (see Fig 5,15). Assume that species $A$ is slightly soluble in $B$. Obtain the physical formulation of the simplified model.

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01:14

Problem 32

Saturated vapor at $100{ }^{\circ} \mathrm{C}$ flows inside a horizontal tube cooled at the external surface. A pool of the liquid water exists at the bottom of the tube and flows with a velocity of $0.5 \mathrm{~m} / \mathrm{s}$. Determine the critical velocity of the vapor when waves appear on the liquid-vapor interface.

Naman Kumar

Numerade Educator

Problem 33

In a stratified, horizontal, two-phase flow system, the saturated water vapor flows above the saturated liquid water. The pressure of the two-phase system is at 5 bar. What is the most dangerous wavelength, $\lambda_D$ ?

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