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Chapter 6: Problem 73

Dry air at \(32^{\circ} \mathrm{C}\) flows over a wetted (water) plate of \(0.2 \mathrm{~m}^{2}\) area. The average convection coefficient is \(\bar{h}=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the heater power required to maintain the plate at a temperature of \(27^{\circ} \mathrm{C}\) is \(432 \mathrm{~W}\). Estimate the power required to maintain the wetted plate at a temperature of \(37^{\circ} \mathrm{C}\) in dry air at \(32^{\circ} \mathrm{C}\) if the convection coefficients remain unchanged.

Short Answer

Expert verified
The power required to maintain the wetted plate at a temperature of \(37^\circ C\) in dry air at \(32^\circ C\) is 0 W, since the plate is already warmer than the surrounding air and the heater is not necessary.

Step by step solution

01

Understand the relationship between power, convection coefficient, and temperature difference

The power required to maintain a certain temperature is directly proportional to the temperature difference between the plate and the dry air, and the average convection coefficient. The relationship can be described using the following equation: \[P = \bar{h} \cdot A \cdot \Delta T\] Where, - \(P\) is the power required (in W) - \(\bar{h}\) is the average convection coefficient (in \(W / m^2 \cdot K\)) - \(A\) is the wetted plate area (in \(m^2\)) - \(\Delta T\) is the temperature difference between the plate and the dry air (in \(^\circ C\))
02

Calculate the temperature difference for the initial condition

For the first case, we need to calculate the temperature difference between the plate and the dry air when the heater is used to maintain the plate at a temperature of \(27^\circ C\). \[\Delta T_1 = T_{p1} - T_{air}\] Where: - \(\Delta T_1\) is the initial temperature difference (in \(^\circ C\)) - \(T_{p1} = 27^\circ C\) is the initial plate temperature - \(T_{air} = 32^\circ C\) is the dry air temperature \[\Delta T_1 = 27^\circ C - 32^\circ C = -5^\circ C\]
03

Calculate the power ratio based on the new temperature difference

We need to find the power required to maintain the plate at a temperature of \(37^\circ C\). First, let's calculate the new temperature difference. \[\Delta T_2 = T_{p2} - T_{air}\] Where: - \(\Delta T_2\) is the new temperature difference (in \(^\circ C\)) - \(T_{p2} = 37^\circ C\) is the new plate temperature \[\Delta T_2 = 37^\circ C - 32^\circ C = 5^\circ C\] Now we find the ratio of the initial and final temperature differences: \[\frac{\Delta T_2}{\Delta T_1} = \frac{5^\circ C}{-5^\circ C} = -1\]
04

Find the power required to maintain the plate at the new temperature

Using the power equation and the fact that power is directly proportional to the temperature difference, we can find the power required to maintain the plate at the new temperature. \[P_2 = P_1 \cdot \frac{\Delta T_2}{\Delta T_1}\] Where: - \(P_1 = 432 W\) is the initial power - \(P_2\) is the power required to maintain the plate at \(37^\circ C\) (in W) \[P_2 = 432 W \cdot (-1) = -432 W\] However, power cannot be negative. In this case, the negative sign indicates that the power required to maintain the wetted plate at \(37^\circ C\) is actually not needed, since the plate is already warmer than the surrounding air. The heater is not necessary, and the power required is therefore 0 W.

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Key Concepts

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

Convection Coefficient
The convection coefficient, often represented by the symbol \( \bar{h} \), is a crucial parameter in the study of convective heat transfer. It quantifies the effectiveness with which heat is transferred from a surface to a fluid moving over it, or vice versa. This coefficient depends on the type of fluid, its velocity, and the properties of the surface in question. In easier terms, it is somewhat like a rating of how good a scenario is at transferring heat through convection.

Convection can either be natural, arising from density differences caused by temperature variations in the fluid, or forced, where the fluid flow is generated by external means like fans or pumps. In the exercise, the convection is presumably forced as it involves air flowing over a plate. Hence, the convection coefficient reflects how efficiently the air can carry away or bring heat to the plate. The higher the convection coefficient, the more efficient the heat transfer process becomes. Thus, knowing the value of \( \bar{h} \) is indispensable for engineers and scientists when designing systems for heating or cooling.
Temperature Difference
Temperature difference, denoted as \( \Delta T \), is the driving force behind heat transfer by convection. It is the difference in temperature between the surface in question, such as our water-wetted plate, and the surrounding fluid鈥攄ry air in our scenario. The heat transfer from the plate to the air (or vice versa) will continue until thermal equilibrium is reached, meaning that the temperature on both sides evens out.

Understanding the temperature difference is crucial as it directly impacts the heat transfer rate. A higher temperature difference usually translates to a greater heat transfer rate, assuming all other factors remain constant. In the exercise, the initial negative temperature difference of \( -5^\circ C\) suggests that the plate is cooler than the surrounding air, and thus heat is being supplied to the plate. On the other hand, a positive temperature difference in the second scenario indicates that the plate is now hotter than the surrounding air.
Heat Transfer Power Calculation
Heat transfer power calculation allows us to determine the amount of power required to maintain a temperature difference between a surface and the surrounding fluid. Power in this context refers to the rate at which energy must be added or removed to keep the surface at a specific temperature. The formula used to calculate this power is:

P = \bar{h} \cdot A \cdot \Delta T
where \(P\) is the power in watts (W), \(\bar{h}\) is the average convection coefficient in \(W/m^2\cdot K\), \(A\) is the area of the surface in square meters (\(m^2\)), and \(\Delta T\) is the temperature difference in degrees Celsius (\(^\circ C\)).

In our textbook exercise, this calculation is performed twice: once for the initial conditions and once for the new conditions. It's important to emphasize that while the negative result \( -432 W\) in Step 4 highlights a misinterpretation, this does not mean power is unnecessary; it simply indicates that no heating is needed as the plate is already at a higher temperature than the air. Always remember, actual physical quantities like power cannot be negative; they indicate a reversal of the heating or cooling direction.

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Most popular questions from this chapter

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 \(V=20 \mathrm{~m} / \mathrm{s}\) ?

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 \(x\) ?

Experiments have been conducted to determine local heat transfer coefficients for flow perpendicular to a long, isothermal bar of rectangular cross section. The bar is of width \(c\) parallel to the flow, and height \(d\) normal to the flow. For Reynolds numbers in the range \(10^{4} \leq R_{d} \leq 5 \times 10^{4}\), the face-averaged Nusselt numbers are well correlated by an expression of the form The values of \(C\) and \(m\) for the front face, side faces, and back face of the rectangular rod are found to be the following: \begin{tabular}{llll} \hline Face & cld & \(\boldsymbol{C}\) & \(\boldsymbol{m}\) \\ \hline Front & \(0.33 \leq\) cld \(51.33\) & \(0.674\) & \(1 / 2\) \\ Side & \(0.33\) & \(0.153\) & \(2 / 3\) \\ Side & \(1.33\) & \(0.107\) & \(2 / 3\) \\ Back & \(0.33\) & \(0.174\) & \(2 / 3\) \\ Back & \(1.33\) & \(0.153\) & \(2 / 3\) \\ \hline \end{tabular} Determine the value of the average heat transfer coefficient for the entire exposed surface (that is, averaged over all four faces) of a \(c=40\)-mm-wide, \(d=30\)-mm-tall rectangular rod. The rod is exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{x}=300 \mathrm{~K}\). Provide a plausible explanation of the relative values of the face-averaged heat transfer coefficients on the front, side, and back faces.

For flow over a flat plate of length \(L\), the local heat transfer coefficient \(h_{x}\) is known to vary as \(x^{-1 / 2}\), where \(x\) is the distance from the leading edge of the plate. What is the ratio of the average Nusselt number for the entire plate \(\left(\overline{N u}_{L}\right)\) to the local Nusselt number at \(x=L\left(N u_{L}\right)\) ?

The defroster of an automobile functions by discharging warm air on the inner surface of the windshield. To prevent condensation of water vapor on the surface, the temperature of the air and the surface convection coefficient \(\left(T_{\infty, j}, \overline{h_{i}}\right)\) must be large enough to maintain a surface temperature \(T_{s i}\) that is at least as high as the dewpoint \(\left(T_{s, i} \geq T_{d \mathrm{p}}\right)\). Consider a windshield of length \(L=800 \mathrm{~mm}\) and thickness \(t=6 \mathrm{~mm}\) and driving conditions for which the vehicle moves at a velocity of \(V=70 \mathrm{mph}\) in ambient air at \(T_{\infty \rho}=-15^{\circ} \mathrm{C}\). From laboratory experiments performed on a model of the vehicle, the average convection coefficient on the outer surface of the windshield is known to be correlated by an expression of the form \(\overline{N_{L}}=0.030 \operatorname{Re}_{L}^{0.8} \operatorname{Pr}^{1 / 3}\), where \(R e_{L}=V L \nu\). Air properties may be approximated as \(k=0.023 \mathrm{~W} / \mathrm{m}=\mathrm{K}\), \(v=12.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\), and \(\operatorname{Pr}=0.71\). If \(T_{d p}=10^{\circ} \mathrm{C}\) and \(T_{m, j}=50^{\circ} \mathrm{C}\), what is the smallest value of \(\bar{h}_{j}\) required to prevent condensation on the inner surface?
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