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Chapter 6: Problem 81

An industrial process involves evaporation of a thin water film from a contoured surface by heating it from below and forcing air across it. Laboratory measurements for this surface have provided the following heat transfer correlation: $$ \overline{N u_{L}}=0.43 R e_{L}^{0.58} P r^{0.4} $$ The air flowing over the surface has a temperature of \(290 \mathrm{~K}\), a velocity of \(10 \mathrm{~m} / \mathrm{s}\), and is completely dry \(\left(\phi_{\infty}=0\right)\). The surface has a length of \(1 \mathrm{~m}\) and a surface area of \(1 \mathrm{~m}^{2}\). Just enough energy is supplied to maintain its steady-state temperature at \(310 \mathrm{~K}\). (a) Determine the heat transfer coefficient and the rate at which the surface loses heat by convection. (b) Determine the mass transfer coefficient and the evaporation rate \((\mathrm{kg} / \mathrm{h})\) of the water on the surface. (c) Determine the rate at which heat must be supplied to the surface for these conditions.

Short Answer

Expert verified
(a) The heat transfer coefficient, h, is calculated using the heat transfer correlation, Reynolds number, Prandtl number, Nusselt number, and air properties. The rate of heat transfer by convection is given by \(q = h A \Delta T\). (b) The mass transfer coefficient, \(k_m\), is calculated from the Sherwood number, which is derived using the heat-mass transfer analogy. The evaporation rate is given by \(q_m = k_m A P_{w_0}\), where \(P_{w_0}\) is the partial pressure of water vapor at the surface. (c) The rate of heat supplied to the surface, Q, is calculated as \(Q = q + L_v q_m\), where \(L_v\) is the latent heat of vaporization.

Step by step solution

01

Calculate Reynolds number

The Reynolds number can be calculated using the formula: \(Re_L = \frac{\rho V L}{\mu}\) where Re_L = Reynolds number 蟻 = air density (calculate using the ideal gas law) V = air velocity (given 10 m/s) L = characteristic length (given 1 m) 渭 = air dynamic viscosity (use standard air properties) To calculate air density and dynamic viscosity, we can use the air temperature given in the problem (290 K) and consult standard air properties.
02

Calculate Prandtl number

The Prandtl number can be calculated using the formula: \(Pr = \frac{c_p \mu}{k}\) where Pr = Prandtl number c_p = specific heat of air (use standard air properties) 渭 = air dynamic viscosity (from Step 1) k = thermal conductivity of air (use standard air properties) Calculate the Prandtl number for the given air properties and temperature.
03

Calculate Nusselt number and heat transfer coefficient

Using the heat transfer correlation given in the problem: \(\overline{N u_{L}}=0.43 Re_{L}^{0.58} Pr^{0.4}\) Substitute the calculated Reynolds number Re_L and the calculated Prandtl number Pr in the given equation and calculate the Nusselt number, \( Nu_L\). Now, the heat transfer coefficient can be calculated as: \(h = \frac{Nu_L k}{L}\) Calculate the heat transfer coefficient, h, using the calculated Nusselt number, thermal conductivity of air k, and characteristic length L = 1 m.
04

Calculate rate of heat transfer by convection

The rate of heat transfer by convection can be calculated as: \(q = h A \Delta T\) where q = rate of heat transfer by convection h = heat transfer coefficient (from Step 3) A = surface area (given 1 m虏) 螖T = difference in temperature between the air and the surface Calculate the rate of heat transfer by convection using the calculated heat transfer coefficient, h, and the given surface-area and temperature difference.
05

Calculate mass transfer coefficient and evaporation rate

Using the heat-mass transfer analogy: \(Nu_L = \frac{hL}{k} = \frac{Sh L}{D}\) where Sh = Sherwood number L = characteristic length (1 m) D = mass diffusivity of water vapor in air (use standard values) Calculate the mass transfer coefficient k_m as: \(k_m = \frac{Sh D}{L}\) Now, with the mass transfer coefficient, the evaporation rate can be calculated as: \(q_m = k_m A P_{w_0}\) where q_m = evaporation rate (kg/h) k_m = mass transfer coefficient (calculated above) A = surface area (1 m虏) \(P_{w_0}\) = partial pressure of water vapor at the surface (calculate using the given surface temperature) Calculate the evaporation rate, q_m, using the calculated mass transfer coefficient and surface area.
06

Calculate the rate of heat supplied to the surface

With the calculated evaporation rate and heat transfer by convection, we can calculate the rate of heat supplied as: \(Q = q + L_v q_m\) where Q = rate of heat supplied q = rate of heat transfer by convection (from Step 4) \(L_v\) = latent heat of vaporization of water (use standard values) \(q_m\) = evaporation rate (from Step 5) Calculate the rate of heat supplied to the surface, Q, using the calculated values in the equation above.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reynolds Number
The Reynolds Number (Re_L) is an important dimensionless quantity in fluid mechanics that helps us understand the flow regime of a fluid. It indicates whether the flow of fluid is laminar or turbulent.
Understanding this can help predict patterns of heat and mass transfer in the system, which is essential for designing efficient industrial processes, like the evaporation of water films. To calculate the Reynolds Number, use the formula: \[Re_L = \frac{\rho V L}{\mu}\] where:
  • \( \rho \) is the density of the fluid
  • \( V \) is the velocity of the fluid
  • \( L \) is the characteristic length of the surface along which the fluid flows
  • \( \mu \) is the dynamic viscosity of the fluid
These properties depend on the specific conditions of the system, such as temperature and pressure. In this specific exercise, the air properties at 290 K are used to determine \( \rho \) and \( \mu \). Understanding the value of the Reynolds Number allows engineers to adjust process conditions for better efficiency and control.
Prandtl Number
The Prandtl Number (Pr) is another dimensionless number that plays a critical role in the study of heat transfer in fluid flows. It gives insight into the relative thickness of the thermal and velocity boundary layers. This means it helps us understand how quickly heat diffuses through a fluid compared to the momentum of the fluid itself.

It is calculated using the formula:\[Pr = \frac{c_p \mu}{k}\] where:
  • \( c_p \) is the specific heat capacity at constant pressure of the fluid
  • \( \mu \) is the dynamic viscosity
  • \( k \) is the thermal conductivity
For a process involving air as in this exercise, all of these properties can be determined using standard air tables for the given air temperature of 290 K. By using the Prandtl Number in combination with the Reynolds Number, we can apply empirical correlations, like the one given in the exercise, to evaluate heat transfer coefficients accurately for specific sets of operational conditions.
Mass Transfer
In processes like evaporation, mass transfer is as important as heat transfer because they both occur simultaneously. The study of mass transfer involves understanding how components (like water vapor) move from a surface into the surrounding air, influenced by concentration differences and environmental conditions.The mass transfer coefficient (k_m) is used to estimate the evaporation rate. It is derived using an analogy between heat and mass transfer described by empirical correlations that incorporate the Sherwood Number (Sh) as follows:\[Nu_L = \frac{hL}{k} = \frac{Sh L}{D}\]In this relation:
  • \( Sh \) is the Sherwood Number, functionally similar to the Nusselt Number used in heat transfer
  • \( L \) is the characteristic length once more
  • \( D \) is the diffusivity of the species in the medium
The evaporation rate, \( q_m \), then depends on:\[ q_m = k_m A P_{w_0} \]where:
  • \( A \) is the area of the surface
  • \( P_{w_0} \) is the water vapor partial pressure at the surface
Knowing the mass transfer coefficient allows engineers to predict and optimize the evaporation rate, ensuring efficient evaporation and energy use in the process.

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Most popular questions from this chapter

For laminar flow over a flat plate, the local heat transfer coefficient \(h_{x}\) is known to vary as \(x^{-1 / 2}\), where \(x\) is the distance from the leading edge \((x=0)\) of the plate. What is the ratio of the average coefficient between the leading edge and some location \(x\) on the plate to the local coefficient at \(x\) ?

As a means of preventing ice formation on the wings of a small, private aircraft, it is proposed that electric resistance heating elements be installed within the wings. To determine representative power requirements, consider nominal flight conditions for which the plane moves at \(100 \mathrm{~m} / \mathrm{s}\) in air that is at a temperature of \(-23^{\circ} \mathrm{C}\). If the characteristic length of the airfoil is \(L=2 \mathrm{~m}\) and wind tunnel measurements indicate an average friction coefficient of \(\bar{C}_{f}=0.0025\) for the nominal conditions, what is the average heat flux needed to maintain a surface temperature of \(T_{s}=5^{\circ} \mathrm{C}\) ?

A fan that can provide air speeds up to \(50 \mathrm{~m} / \mathrm{s}\) is to be used in a low-speed wind tunnel with atmospheric air at \(25^{\circ} \mathrm{C}\). If one wishes to use the wind tunnel to study flatplate boundary layer behavior up to Reynolds numbers of \(R e_{x}=10^{8}\), what is the minimum plate length that should be used? At what distance from the leading edge would transition occur if the critical Reynolds number were \(R e_{x, c}=5 \times 10^{5}\) ?

Air at a free stream temperature of \(T_{a}=20^{\circ} \mathrm{C}\) is in parallel flow over a flat plate of length \(L=5 \mathrm{~m}\) and temperature \(T_{s}=90^{\circ} \mathrm{C}\). However, obstacles placed in the flow intensify mixing with increasing distance \(x\) from the leading edge, and the spatial variation of temperatures measured in the boundary layer is correlated by an expression of the form \(T\left({ }^{\circ} \mathrm{C}\right)=20+70\) \(\exp (-600 x y)\), where \(x\) and \(y\) are in meters. Determine and plot the manner in which the local convection coefficient \(h\) varies with \(x\). Evaluate the average convection coefficient \(\bar{h}\) for the plate.

The heat transfer rate per unit width (normal to the page) from a longitudinal section, \(x_{2}-x_{1}\), can be expressed as \(q_{12}^{\prime}=\bar{h}_{12}\left(x_{2}-x_{1}\right)\left(T_{s}-T_{\infty}\right)\), where \(\bar{h}_{12}\) is the average coefficient for the section of length \(\left(x_{2}-x_{1}\right)\). Consider laminar flow over a flat plate with a uniform temperature \(T_{s}\). The spatial variation of the local convection coefficient is of the form \(h_{x}=C x^{-1 / 2}\), where \(C\) is a constant. (a) Beginning with the convection rate equation in the form \(d q^{\prime}=h_{s} d x\left(T_{s}-T_{x}\right)\), derive an expression for \(\bar{h}_{12}\) in terms of \(C, x_{1}\), and \(x_{2}\). (b) Derive an expression for \(\bar{h}_{12}\) in terms of \(x_{1}, x_{2}\), and the average coefficients \(\bar{h}_{1}\) and \(\bar{h}_{2}\), corresponding to lengths \(x_{1}\) and \(x_{2}\), respectively.
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