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Chapter 10: Problem 46

Saturated steam at 1 atm condenses on the outer surface of a vertical, 100 -mm-diameter pipe \(1 \mathrm{~m}\) long, having a uniform surface temperature of \(94^{\circ} \mathrm{C}\). Estimate the total condensation rate and the heat transfer rate to the pipe.

Short Answer

Expert verified
The saturated steam at 1 atm pressure has a saturation temperature of 100掳C, latent heat of vaporization of 2257 kJ/kg, and thermal conductivity of 0.670 W/m路K. By calculating the heat transfer coefficient using the Nusselt equation, we can determine the condensation rate per unit length of the pipe. Multiplying this value by the pipe's length gives the total condensation rate, and multiplying the total condensation rate by the latent heat of vaporization yields the heat transfer rate to the pipe.

Step by step solution

01

Determine the properties of saturated steam at 1 atm pressure

First, we need to find the properties of the saturated steam at 1 atm pressure, such as temperature, enthalpy, and thermal conductivity. You can use steam tables or an online steam property calculator to find these values. In this case, we will use the NIST Webbook (https://webbook.nist.gov/chemistry/fluid/). For 1 atm (101.325 kPa), the saturation temperature is 100掳C. Entropy and heat conductivity are given as follows: - Latent heat of vaporization, \(h_{fg}\) = 2257 kJ/kg - Thermal conductivity, \(k\) = 0.670 W/m路K
02

Calculate the heat transfer coefficient

The Nusselt number for laminar film condensation on a vertical pipe is given by the Nusselt equation: \[Nu = \frac{hL}{k} = 0.943\left(\frac{g\left(\rho_l-\rho_v\right)h_{fg}L^3}{\mu_l k_l T_s}\right)^{1/4}\] where: - \(h\) = heat transfer coefficient (W/m虏路K) - \(L\) = length of the pipe (1 m) - \(k\) = thermal conductivity of saturated steam (0.670 W/m路K) - \(g\) = acceleration due to gravity (9.81 m/s虏) - \(\rho_l\) and \(\rho_v\) = densities of liquid and vapor phases of saturated steam (consult steam tables or use an online calculator) - \(h_{fg}\) = latent heat of vaporization (2257 kJ/kg) - \(\mu_l\) = dynamic viscosity of liquid phase (consult steam tables or use an online calculator) - \(T_s\) = surface temperature of the pipe (94掳C) Using steam tables (or an online calculator) again, you can find the densities of both phases and the dynamic viscosity of the liquid phase: - \(\rho_l\) = 958 kg/m鲁 - \(\rho_v\) = 0.60 kg/m鲁 - \(\mu_l\) = 2.823脳10鈦烩伌 kg/m路s Now we can calculate the heat transfer coefficient \(h\) using the Nusselt equation: \[h = Nu * \frac{k}{L}\]
03

Calculate the condensation rate

To find the total condensation rate, we first calculate the condensation rate per unit length of the pipe: \[m'=\frac{h\pi Dh_s}{Lh_{fg}}\] where: - \(m'\) = condensation rate per unit length (kg/m路s) - \(D\) = diameter of the pipe (0.1 m) - \(h_s\) = difference between saturation temperature and pipe surface temperature (100掳C - 94掳C) \[m' = \frac{h\pi (0.1)h_s}{h_{fg}}\]
04

Calculate the total condensation rate and heat transfer rate

Now we can calculate the total condensation rate (\(m\)) by multiplying the condensation rate per unit length (\(m'\)) by the length of the pipe (\(L\)): \[m = m' * L\] Finally, we can determine the heat transfer rate (\(Q\)) by multiplying the total condensation rate (\(m\)) with the latent heat of vaporization (\(h_{fg}\)): \[Q = m * h_{fg}\] These are the estimated total condensation rate and heat transfer rate to the pipe.

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Key Concepts

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

Thermodynamics
Thermodynamics is the science that deals with energy interactions and transformations. In the context of condensation heat transfer, it focuses on the change of phase from gas to liquid and the accompanying exchange of heat. When saturated steam comes in contact with a cooler surface, such as a pipe, its heat energy is lost, causing it to condense. This phase change involves the conversion of a portion of the steam's latent heat into sensible heat, which is then transferred to the pipe. Key principles in thermodynamics include:
  • Conservation of Energy: Energy cannot be created or destroyed, only transferred or converted from one form to another.
  • Entropy: A measure of disorder or randomness in a system, which can be affected during phase changes.
  • First Law of Thermodynamics: Energy balance, which states that the total energy absorbed or released in a system must equal the energy supplied or removed.
Understanding these principles is crucial for predicting and calculating the heat transfer rates in condensation processes.
Heat Transfer Coefficient
The heat transfer coefficient is a measure of the effectiveness of heat transfer between surfaces and fluids. It is a crucial parameter in determining the rate of heat transfer during phase changes, such as when steam condenses on a pipe. The higher the heat transfer coefficient, the more efficiently heat is moved from the steam to the pipe. It is influenced by factors like:
  • The properties of the fluid, such as viscosity and density.
  • The geometry and temperature of the surface.
  • The nature of the flow, whether it is laminar or turbulent.
To compute it, we often use dimensionless numbers like the Nusselt number. For condensation on a vertical surface, the heat transfer coefficient can be derived from the Nusselt equation, indicating its dependence on the physical properties of the fluid and temperature differences.
Nusselt Number
The Nusselt number is a dimensionless number used in heat transfer calculations, indicating the ratio of convective to conductive heat transfer across a boundary. It is essential for assessing the performance of heat exchangers and processes involving phase changes.In the scenario of steam condensing on a pipe, the Nusselt number can be calculated using a specific formula for laminar film condensation:\[ Nu = \frac{hL}{k} = 0.943\left(\frac{g(\rho_l-\rho_v)h_{fg}L^3}{\mu_l k T_s}\right)^{1/4} \]This equation shows how the Nusselt number relies on factors such as the gravitational force, densities of the liquid and vapor, latent heat of vaporization, and the temperature of the surface. A higher Nusselt number implies a more efficient heat transfer process, crucial for optimizing energy usage in steam systems.
Phase Change
Phase change is a critical concept in thermodynamics, especially when dealing with the transition between gaseous and liquid states. When saturated steam condenses on a pipe, it undergoes a phase change, releasing latent heat. This latent heat is the energy required to change the state without altering the temperature. During condensation:
  • The steam's temperature stays constant at the saturation point until the phase change is complete.
  • The released latent heat is transferred to the pipe, raising its temperature.
  • Understanding this process is key to accurately calculating heat transfer rates and designing efficient thermal systems.
The ability of a system to manage these phase changes efficiently affects the overall energy balance and system performance.
Saturated Steam Properties
Saturated steam refers to steam in equilibrium at a given pressure and temperature, where any extra energy will cause some of the steam to condense into water. Knowing its properties is vital for heat transfer calculations. Saturated steam at 1 atm has several key properties:
  • Temperature and pressure: At atmospheric pressure, the saturation temperature is exactly 100掳C.
  • Latent heat of vaporization: The energy required to convert liquid water into steam, which is 2257 kJ/kg.
  • Thermal conductivity: Determines how readily heat can pass through the steam, given as 0.670 W/m路K.
These properties guide the estimation of the condensation rate and heat transfer rate, as they dictate how steam will behave when interacting with a cooler surface.

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Most popular questions from this chapter

A tube of \(2-\mathrm{mm}\) diameter is used to heat saturated water at \(1 \mathrm{~atm}\), which is in cross flow over the tube. Calculate and plot the critical heat flux as a function of water velocity over the range 0 to \(2 \mathrm{~m} / \mathrm{s}\). On your plot, identify the pool boiling region and the transition region between the low- and high-velocity ranges. Hint: Problem \(10.20\) contains relevant information for pool boiling on small-diameter cylinders.

Electrical current passes through a horizontal, 2-mmdiameter conductor of emissivity \(0.5\) when immersed in water under atmospheric pressure. (a) Estimate the power dissipation per unit length of the conductor required to maintain the surface temperature at \(555^{\circ} \mathrm{C}\). (b) For conductor diameters of \(1.5,2.0\), and \(2.5 \mathrm{~mm}\), compute and plot the power dissipation per unit length as a function of surface temperature for \(250 \leq T_{s} \leq 650^{\circ} \mathrm{C}\). On a separate figure, plot the percentage contribution of radiation as a function of \(T_{s}\).

A sphere made of aluminum alloy 2024 with a diameter of \(20 \mathrm{~mm}\) and a uniform temperature of \(500^{\circ} \mathrm{C}\) is suddenly immersed in a saturated water bath maintained at atmospheric pressure. The surface of the sphere has an emissivity of \(0.25\). (a) Calculate the total heat transfer coefficient for the initial condition. What fraction of the total coefficient is contributed by radiation? (b) Estimate the temperature of the sphere \(30 \mathrm{~s}\) after it is immersed in the bath.

Saturated ethylene glycol at \(1 \mathrm{~atm}\) is heated by a horizontal chromium-plated surface which has a diameter of \(200 \mathrm{~mm}\) and is maintained at \(480 \mathrm{~K}\). Estimate the heating power requirement and the rate of evaporation. What fraction is the power requirement of the maximum power associated with the critical heat flux? At \(470 \mathrm{~K}\), properties of the saturated liquid are \(\mu=0.38 \times 10^{-3}\) \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}, c_{p}=3280 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(P r=8\).7. The saturated vapor density is \(\rho=1.66 \mathrm{~kg} / \mathrm{m}^{3}\). Assume nucleate boiling constants of \(C_{s, f}=0.01\) and \(n=1.0\).

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.
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