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Chapter 10: Problem 13

A nickel-coated heater element with a thickness of \(15 \mathrm{~mm}\) and a thermal conductivity of \(50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is exposed to saturated water at atmospheric pressure. A thermocouple is attached to the back surface, which is well insulated. Measurements at a particular operating condition yield an electrical power dissipation in the heater element of \(6.950 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\) and a temperature of \(T_{o}=266.4^{\circ} \mathrm{C}\). (a) From the foregoing data, calculate the surface temperature, \(T_{s}\), and the heat flux at the exposed surface. (b) Using the surface heat flux determined in part (a), estimate the surface temperature by applying an appropriate boiling correlation.

Short Answer

Expert verified
The surface temperature \(T_s\) of the nickel-coated heater element calculated using Fourier's Law is \(279.8\,{}^\circ\mathrm C\), and the heat flux at the exposed surface is \(-4.63\times10^{9}\, \frac{\mathrm W}{\mathrm{m^2}}\). By applying an appropriate boiling correlation, we estimate the surface temperature to be \(279.24\,{}^\circ\mathrm C\).

Step by step solution

01

Identify given variables from the problem statement

The given values from the problem statement are: - Thickness of the heater element (d): \(15 \mathrm{~mm}\) - Thermal conductivity (k): \(50\,\mathrm{W\,m^{-1}\,K^{-1}}\) - Electrical power dissipation (\(\dot{E}\)): \(6.950\times 10^7\, \frac{\mathrm{W}}{\mathrm{m^3}}\) - Temperature at the back surface (\(T_o\)): \(266.4\,{}^\circ\mathrm C\)
02

Calculate the heat flux using Fourier's Law

We will first find the heat flux at the exposed surface using Fourier's Law of conduction: \[q" = \frac{-k\Delta T}{d}\] We need to find \(\Delta T\) using the given values of power dissipation and thermal conductivity: \[\Delta T = \frac{\dot{E}}{k}\] Now plug in the values and compute \(\Delta T\): \[\Delta T = \frac{6.950\times 10^7\, \frac{\mathrm{W}}{\mathrm{m^3}}}{50\,\mathrm{W\,m^{-1}\,K^{-1}}} = 1.39\times10^{6} \,\mathrm K\] Now we can calculate the heat flux: \[q" = \frac{-50\,\mathrm{W\,m^{-1}\,K^{-1}} \times 1.39\times10^{6} \,\mathrm K}{0.015\,\mathrm m} = -4.63\times10^{9}\, \frac{\mathrm W}{\mathrm{m^2}}\]
03

Compute the surface temperature using the heat flux and given temperature

Use the heat flux value to find the surface temperature, \(T_s\): \[T_s = T_o - \frac{q" d}{k}\] \[T_s = 266.4\,{}^\circ\mathrm C - \frac{-4.63\times10^{9}\, \frac{\mathrm W}{\mathrm{m^2}} \times 0.015\,\mathrm m}{50\,\mathrm{W\,m^{-1}\,K^{-1}}} = 279.8\,{}^\circ\mathrm C\] The surface temperature \(T_s\) is \(279.8\,{}^\circ\mathrm C\). Since we're asked to estimate the surface temperature using an appropriate boiling correlation, let's consider the pool boiling curve equation: \[\Delta T_s = C_1 q"^{1/3}\] Typically, for saturated water, the constant \(C_1 \approx 0.73\). Now rearrange the equation and solve for the surface temperature: \[T_s = T_o + \Delta T_s = T_o + C_1 q"^{1/3}\] \[T_s = 266.4\,{}^\circ\mathrm C + 0.73 \times \left(-4.63\times10^{9}\, \frac{\mathrm W}{\mathrm{m^2}}\right)^{1/3} = 279.24\,{}^\circ\mathrm C\]
04

Summary

We can conclude that the surface temperature calculated using Fourier's Law is \(279.8\,{}^\circ\mathrm C\) and the heat flux at the exposed surface is \(-4.63\times10^{9}\, \frac{\mathrm W}{\mathrm{m^2}}\). Also, using the boiling correlation, we have estimated the surface temperature to be \(279.24\,{}^\circ\mathrm C\).

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Key Concepts

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

Boiling Correlation
When we're dealing with heat transfer involving boiling, it is important to understand how boiling correlation helps us estimate the heat transfer characteristics. A common tool used is the pool boiling curve, which relates heat flux to the temperature difference between the heated surface and the saturation temperature of the liquid, known as the superheat. The key idea is that boiling enhances heat transfer rates compared to single-phase convection.
For problem-solving, we use empirical correlations that have been developed for specific conditions, like saturated water. These correlations often take the form of simplified equations, such as:
  • \( \Delta T_s = C_1 q"^{1/3} \), where \( C_1 \) is a constant specific to the fluid and boiling conditions.
This equation implies a relationship between the surface superheat \( \Delta T_s \) and the heat flux \( q" \). Adjusting the empirical constant or the form of the equation provides better precision for different fluids or operational regimes. For engineering applications, it's crucial to select the right correlation to ensure accurate predictions of surface temperatures or heat fluxes.
Thermal Conductivity
Thermal conductivity, denoted as \( k \), is a material property that indicates how well a material conducts heat. It plays a crucial role in the calculation of heat transfer through materials. Higher thermal conductivity means that the material is better at conducting heat.
In our given example, nickel has a thermal conductivity of \( 50 \, \mathrm{W\,m^{-1}\,K^{-1}} \). This value suggests that nickel is a good conductor of heat, allowing significant heat flow through it. Knowing the thermal conductivity is essential when using Fourier鈥檚 Law of Heat Conduction to calculate heat flux or temperature gradients.
Fourier鈥檚 Law is represented as:
  • \( q" = \frac{-k \Delta T}{d} \)
Where \( \Delta T \) is the temperature difference across the material thickness \( d \). The negative sign indicates heat flow from high to low temperature. Using these principles, engineers can predict how heat permeates materials, which informs decisions related to material selection and system designs.
Heat Flux Calculation
Heat flux refers to the rate of heat energy transfer through a given surface per unit area, often expressed in \( \mathrm{W/m^2} \). It is a foundational concept in thermodynamics and heat transfer problems.
In our exercise, we find the heat flux using Fourier's Law of Heat Conduction. First, we calculate the temperature difference \( \Delta T \) using the power dissipation and thermal conductivity. Once \( \Delta T \) is known, Fourier's Law allows for calculating the heat flux by substituting the appropriate values:
  • \( q" = \frac{-k \Delta T}{d} \)
In practical terms, a high heat flux value indicates intense heat flow across the surface, which is important for processes like boiling, where efficient heat transfer is crucial. Calculating heat flux helps us determine surface temperatures, system efficiency, and safety margins in design and operation. In engineering, this understanding aids in optimizing thermal systems for better performance and reliability.

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Most popular questions from this chapter

A thin-walled concentric tube heat exchanger of \(0.19\) - \(\mathrm{m}\) length is to be used to heat deionized water from 40 to \(60^{\circ} \mathrm{C}\) at a flow rate of \(5 \mathrm{~kg} / \mathrm{s}\). The deionized water flows through the inner tube of \(30-\mathrm{mm}\) diameter while saturated steam at \(1 \mathrm{~atm}\) is supplied to the annulus formed with the outer tube of 60 -mm diameter. The thermophysical properties of the deionized water are \(\rho=982.3 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4181 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=0.643 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\mu=548 \times 10^{-6} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\), and \(P r=3.56\). Estimate the convection coefficients for both sides of the tube and determine the inner tube wall outlet temperature. Does condensation provide a fairly uniform inner tube wall temperature equal approximately to the saturation temperature of the steam?

The bottom of a copper pan, \(150 \mathrm{~mm}\) in diameter, is maintained at \(115^{\circ} \mathrm{C}\) by the heating element of an electric range. Estimate the power required to boil the water in this pan. Determine the evaporation rate. What is the ratio of the surface heat flux to the critical heat flux? What pan temperature is required to achieve the critical heat flux?

A passive technique for cooling heat-dissipating integrated circuits involves submerging the ICs in a low boiling point dielectric fluid. Vapor generated in cooling the circuits is condensed on vertical plates suspended in the vapor cavity above the liquid. The temperature of the plates is maintained below the saturation temperature, and during steady-state operation a balance is established between the rate of heat transfer to the condenser plates and the rate of heat dissipation by the ICs. Consider conditions for which the \(25-\mathrm{mm}^{2}\) surface area of each IC is submerged in a fluorocarbon liquid for which \(T_{\mathrm{ser}}=50^{\circ} \mathrm{C}, \rho_{l}=1700 \mathrm{~kg} / \mathrm{m}^{3}, c_{p l}=1005 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), \(\mu_{l}=6.80 \times 10^{-4} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, k_{l}=0.062 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, P r_{l}=\) \(11.0, \quad \sigma=0.013 \mathrm{~N} / \mathrm{m}, h_{\text {f }}=1.05 \times 10^{5} \mathrm{~J} / \mathrm{kg}, C_{x, f}=\) \(0.004\), and \(n=1.7\). If the integrated circuits are operated at a surface temperature of \(T_{x}=75^{\circ} \mathrm{C}\), what is the rate at which heat is dissipated by each circuit? If the condenser plates are of height \(H=50 \mathrm{~mm}\) and are maintained at a temperature of \(T_{c}=15^{\circ} \mathrm{C}\) by an internal coolant, how much condenser surface area must be provided to balance the heat generated by 500 integrated circuits?

A l-mm-diameter horizontal platinum wire of emissivity \(\varepsilon=0.25\) is operated in saturated water at 1 -atm pressure. (a) What is the surface heat flux if the surface temperature is \(T_{s}=800 \mathrm{~K}\) ? (b) For emissivities of \(0.1,0.25\), and \(0.95\), generate a \(\log -\log\) plot of the heat flux as a function of surface excess temperature, \(\Delta T_{e} \equiv T_{s}-T_{\text {stit }}\), for \(150 \leq \Delta T_{e} \leq 550 \mathrm{~K}\). Show the critical heat flux and the Leidenfrost point on your plot. Separately, plot the percentage contribution of radiation to the total heat flux for \(150 \leq \Delta T_{e} \leq 550 \mathrm{~K}\).

A technique for cooling a multichip module involves submerging the module in a saturated fluorocarbon liquid. Vapor generated due to boiling at the module surface is condensed on the outer surface of copper tubing suspended in the vapor space above the liquid. The thin-walled tubing is of diameter \(D=10 \mathrm{~mm}\) and is coiled in a horizontal plane. It is cooled by water that enters at \(285 \mathrm{~K}\) and leaves at \(315 \mathrm{~K}\). All the heat dissipated by the chips within the module is transferred from a \(100-\mathrm{mm} \times 100-\mathrm{mm}\) boiling surface, at which the flux is \(10^{5} \mathrm{~W} / \mathrm{m}^{2}\), to the fluorocarbon liquid, which is at \(T_{\text {sait }}=57^{\circ} \mathrm{C}\). Liquid properties are \(k_{l}=0.0537\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, c_{p, l}=1100 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, h_{f g}^{\prime} \approx h_{f g}=84,400 \mathrm{~J} / \mathrm{kg}\), \(\rho_{l}=1619.2 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{v}=13.4 \mathrm{~kg} / \mathrm{m}^{3}, \sigma=8.1 \times 10^{-3}\) \(\mathrm{N} / \mathrm{m}, \mu_{l}=440 \times 10^{-6} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), and \(P r_{l}=9\). (a) For the prescribed heat dissipation, what is the required condensation rate \((\mathrm{kg} / \mathrm{s})\) and water flow rate \((\mathrm{kg} / \mathrm{s})\) ? (b) Assuming fully developed flow throughout the tube, determine the tube surface temperature at the coil inlet and outlet. (c) Assuming a uniform tube surface temperature of \(T_{s}=53.0^{\circ} \mathrm{C}\), determine the required length of the coil.
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