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Chapter 6: Problem 59

An object of irregular shape has a characteristic length of \(L=1 \mathrm{~m}\) and is maintained at a uniform surface temperature of \(T_{s}=325 \mathrm{~K}\). It is suspended in an airstream that is at atmospheric pressure \((p=1 \mathrm{~atm})\) and has a velocity of \(V=100 \mathrm{~m} / \mathrm{s}\) and a temperature of \(T_{x}=275 \mathrm{~K}\). The average heat flux from the surface to the air is \(12,000 \mathrm{~W} / \mathrm{m}^{2}\). Referring to the foregoing situation as case 1 , consider the following cases and determine whether conditions are analogous to those of case 1. Each case involves an object of the same shape, which is suspended in an airstream in the same manner. Where analogous behavior does exist, determine the corresponding value of the average convection coefficient. (a) The values of \(T_{s}, T_{x}\), and \(p\) remain the same, but \(L=2 \mathrm{~m}\) and \(V=50 \mathrm{~m} / \mathrm{s}\). (b) The values of \(T_{x}\) and \(T_{\infty}\) remain the same, but \(L=2 \mathrm{~m}, V=50 \mathrm{~m} / \mathrm{s}\), and \(p=0.2 \mathrm{~atm}\). (c) The surface is coated with a liquid film that evaporates into the air. The entire system is at \(300 \mathrm{~K}\), and the diffusion coefficient for the air-vapor mixture is \(D_{\mathrm{AB}}=1.12 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\). Also, \(L=2 \mathrm{~m}\), \(V=50 \mathrm{~m} / \mathrm{s}\), and \(p=1 \mathrm{~atm}\). (d) The surface is coated with another liquid film for which \(D_{\mathrm{AB}}=1.12 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), and the system is at \(300 \mathrm{~K}\). In this case \(L=2 \mathrm{~m}, V=250 \mathrm{~m} / \mathrm{s}\), and \(p=0.2\) atm.

Short Answer

Expert verified
The behavior of case (a) is analogous to case 1, with an average convection coefficient of 240 W/m虏K. Cases (b), (c), and (d) are not analogous to case 1, and it is not necessary to calculate the average convection coefficient for these cases.

Step by step solution

01

Identify analogous conditions

Since the values of \(T_s\), \(T_x\), and \(p\) remain the same, and only \(L\) and \(V\) are different, the behavior of this case is analogous to case 1.
02

Calculate the average convection coefficient

To calculate the average convection coefficient, use the average heat flux formula: \[ q'' = h(T_s - T_x) \] From case 1, we know that: - The average heat flux (\(q''\)) is \(12,000 \, \mathrm{W/m^2}\); - The surface temperature (\(T_s\)) is \(325 \, \mathrm{K}\); - The air temperature (\(T_x\)) is \(275 \, \mathrm{K}\); Rearrange the formula for the average convection coefficient: \[ h = \frac{q''}{T_s - T_x} = \frac{12,000}{325 - 275} = 240 \, \mathrm{W/m^2K} \] The average convection coefficient for case (a) is 240 W/m虏K. #Case (b)#
03

Identify analogous conditions

Since the values of \(T_x\) and \(T_\infty\) remain the same, and only \(L\), \(V\), and \(p\) are different, the behavior of this case is not analogous to case 1. There is no need to calculate the average convection coefficient in this case. #Case (c)#
04

Identify analogous conditions

In this case, the surface is coated with a liquid film, and the entire system is at a different temperature than case 1. Additionally, there is a diffusion coefficient for the air-vapor mixture. The behavior of this case is not analogous to case 1. There is no need to calculate the average convection coefficient in this case. #Case (d)#
05

Identify analogous conditions

In this case, the surface is coated with another liquid film, and the entire system is at a different temperature than case 1. Additionally, there is a diffusion coefficient for the air-vapor mixture and different values for \(L\), \(V\), and \(p\). The behavior of this case is not analogous to case 1. There is no need to calculate the average convection coefficient in this case. In summary: - The behavior of case (a) is analogous to case 1, with an average convection coefficient of 240 W/m虏K. - Cases (b), (c), and (d) are not analogous to case 1, and it is not necessary to calculate the average convection coefficient for these cases.

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Key Concepts

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

average convection coefficient
In heat transfer, the average convection coefficient, often denoted as \( h \), is a crucial parameter. It quantifies the convective heat transfer between a surface and a fluid flowing over it. To find this coefficient, we often use the formula:
\[ h = \frac{q''}{T_s - T_x} \]
Where:
  • \( q'' \) is the heat flux, or the rate of heat transfer per unit area.
  • \( T_s \) is the surface temperature.
  • \( T_x \) is the fluid temperature.
A higher convection coefficient indicates more efficient heat transfer. In scenarios like case (a), where conditions match with baseline case attributes, the heat flux formula allows the calculation of the average convection coefficient. This drives home the importance of accurately knowing your temperatures and heat flux to solve these problems.
heat flux formula
The heat flux is often calculated using the heat flux formula, an essential tool in solving heat transfer problems. Heat flux, \( q'' \), represents how much heat energy passes through a unit area per unit time and is given the formula:
\[ q'' = h(T_s - T_x) \]
Here, the formula directly ties heat flux to the average convection coefficient and the temperature difference between the surface (\( T_s \)) and the fluid (\( T_x \)). Higher temperature differences or convection coefficients typically yield higher heat flux, illustrating how heat transfer efficiency depends on these parameters. This formula provides a straightforward method to alternately determine unknown variables when the heat context surrounding an object in a fluid is understood.
diffusion coefficient
The diffusion coefficient, \( D_{AB} \), appears in the context of cases where diffusion becomes a relevant factor alongside heat transfer. In the exercise, this coefficient is essential in cases with an evaporating liquid film. This parameter explains how quickly molecules spread through space due to their random thermal motion.
The units of \( D_{AB} \) are typically \( \,\mathrm{m^2/s} \), and it quantifies the ease with which a molecule type \( A \) spreads through type \( B \). In heat and mass transfer problems, understanding how diffusion affects heat transfer helps predict and calculate the transport processes, enhancing the accuracy of predictions and solutions in situations involving phase change.
analogous conditions
"Analogous conditions" in heat transfer refer to situations where cases are considered comparable to a known standard or baseline case. This concept helps determine if parameters like temperature, speed, and pressure mimic a reference case, potentially allowing us to apply the same results, like the convection coefficient, without recalculating from scratch.
When these conditions are met, it suggests that the physics governing heat transfer remains unchanged, despite variations in specific details like geometry or object size. This principle is particularly useful in engineering and physics, facilitating quick assessments and useful approximations of heat transfer behaviors across varying conditions. It's essential to carefully assess each parameter to ensure valid comparisons are drawn between analogous cases.

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Most popular questions from this chapter

An industrial process involves the evaporation of water from a liquid film that forms on a contoured surface. Dry air is passed over the surface, and from laboratory measurements the convection heat transfer correlation is of the form $$ \overline{N_{L}}=0.43 \operatorname{Re}_{L}^{0.58} P r r^{\Omega .4} $$ (a) For an air temperature and velocity of \(27^{\circ} \mathrm{C}\) and \(10 \mathrm{~m} / \mathrm{s}\), respectively, what is the rate of evaporation from a surface of \(1-\mathrm{m}^{2}\) area and characteristic length \(L=1 \mathrm{~m}\) ? Approximate the density of saturated vapor as \(\rho_{A, \text { sat }}=0.0077 \mathrm{~kg} / \mathrm{m}^{3}\). (b) What is the steady-state temperature of the liquid film?

A disk of 20-mm diameter is covered with a water film. Under steady-state conditions, a heater power of \(200 \mathrm{~mW}\) is required to maintain the disk-water film at \(305 \mathrm{~K}\) in dry air at \(295 \mathrm{~K}\) and the observed evaporation rate is \(2.55 \times 10^{-4} \mathrm{~kg} / \mathrm{h}\). (a) Calculate the average mass transfer convection coefficient \(\bar{h}_{s}\) for the evaporation process. (b) Calculate the average heat transfer convection coefficient \(\bar{h}\). (c) Do the values of \(\bar{h}_{\mathrm{m}}\) and \(\bar{h}\) satisfy the heat- mass analogy? (d) If the relative humidity of the ambient air at \(295 \mathrm{~K}\) were increased from 0 (dry) to \(0.50\), but the power supplied to the heater was maintained at \(200 \mathrm{~mW}\), would the evaporation rate increase or decrease? Would the disk temperature increase or decrease?

If laminar flow is induced at the surface of a disk due to rotation about its axis, the local convection coefficient is known to be a constant, \(h=C\), independent of radius. Consider conditions for which a disk of radius \(r_{o}=100 \mathrm{~mm}\) is rotating in stagnant air at \(T_{\infty}=20^{\circ} \mathrm{C}\) and a value of \(C=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) is maintained. If an embedded electric heater maintains a surface temperature of \(T_{x}=50^{\circ} \mathrm{C}\), what is the local heat flux at the top surface of the disk? What is the total electric power requirement? What can you say about the nature of boundary layer development on the disk?

A mist cooler is used to provide relief for a fatigued athlete. Water at \(T_{i}=10^{\circ} \mathrm{C}\) is injected as a mist into a fan airstream with ambient temperature of \(T_{=}=32^{\circ} \mathrm{C}\). The droplet diameters are \(100 \mu \mathrm{m}\). For small droplets the average Nusselt number is correlated by an expression of the form $$ \overline{N u}_{D}=\bar{h} D / k=2 $$ (a) At the initial time, calculate the rate of convection heat transfer to the droplet, the rate of evaporative heat loss, and the rate of change of temperature of the droplet for two values of the relative humidity of the fan airstream, \(\phi_{x}=0.20\) and \(0.95\). Explain what is happening to the droplet in each case. (b) Calculate the steady-state droplet temperature for each of the two relative humidity values in part (a).

An object of irregular shape has a characteristic length of \(L=1 \mathrm{~m}\) and is maintained at a uniform surface temperature of \(T_{s}=400 \mathrm{~K}\). When placed in atmospheric air at a temperature of \(T_{x}=300 \mathrm{~K}\) and moving with a velocity of \(V=100 \mathrm{~m} / \mathrm{s}\), the average heat flux from the surface to the air is \(20,000 \mathrm{~W} / \mathrm{m}^{2}\). If a second object of the same shape, but with a characteristic length of \(L=5 \mathrm{~m}\), is maintained at a surface temperature of \(T_{s}=400 \mathrm{~K}\) and is placed in atmospheric air at \(T_{\infty}=300 \mathrm{~K}\), what will the value of the average convection coefficient be if the air velocity is \(V=20 \mathrm{~m} / \mathrm{s}\) ?
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