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Chapter 8: Problem 3

A \(1.8 \mathrm{~kg} / \mathrm{s}\) supply of saturated steam at \(0.11 \mathrm{MPa}\) is available for heating water in a bottle washing operation at a soft drink bottling plant. What is the maximum flow rate of water that can be heated from \(20^{\circ} \mathrm{C}\) to \(70^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
The maximum flow rate of water is approximately \(18.95 \text{ kg/s}\).

Step by step solution

01

Determine the Heat Required for Heating Water

To calculate the amount of heat required to heat water, we use the formula for specific heat: \[ Q = m_w \cdot C_w \cdot (T_f - T_i) \] where \( Q \) is the heat energy, \( m_w \) is the mass of the water, \( C_w \) is the specific heat capacity of water (\(4.18 \text{ kJ/kg}^\circ \text{C}\)), \( T_f \) is the final temperature, and \( T_i \) is the initial temperature. The temperature change is \(50^{\circ} \text{C}\).
02

Calculate the Heat Supplied by Steam

The heat provided by saturated steam at pressure is given by steam tables. At \(0.11 \text{ MPa}\), the latent heat of vaporization \(h_{fg}\) is approximately \(2201 \text{ kJ/kg}\). The heat energy supplied by the steam is: \[ Q = m_s \cdot h_{fg} \] where \( m_s \) is the mass flow rate of steam, which is \(1.8 \text{ kg/s}\). So, \( Q = 1.8 \cdot 2201 \).
03

Equate Heat Supplied to Heat Required

Set the heat supplied equal to the heat required to find the maximum mass flow rate of water: \[ 1.8 \cdot 2201 = m_w \cdot 4.18 \cdot 50 \].
04

Solve for Mass Flow Rate of Water

Rearrange the equation to solve for \( m_w \): \[ m_w = \frac{1.8 \cdot 2201}{4.18 \cdot 50} \] Calculate \( m_w \).
05

Calculation and Final Answer

Perform the calculations: \[ m_w = \frac{3961.8}{209} \approx 18.95 \text{ kg/s} \]. Therefore, the maximum flow rate of water that can be heated is approximately \(18.95 \text{ kg/s}\).

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Key Concepts

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

Specific Heat Capacity
Specific heat capacity denotes how much heat energy is required to raise the temperature of a certain mass of a substance by one degree Celsius. In simpler terms, it tells you how much energy you need to "warm up" a material. For water, this magic number is quite high: approximately 4.18 kJ/kg掳C. This means water can absorb a lot of heat before it gets noticeably warmer. Think of it like water being a very calorific sponge.

Why is specific heat capacity important?
  • It's crucial for tasks involving heating substances, like in industrial processes or climate control systems.
  • It allows engineers to calculate how much energy is required to achieve certain temperature changes.
Latent Heat of Vaporization
The latent heat of vaporization is a special concept related to heat. It refers to the amount of energy needed to change a substance from a liquid to a gas, without changing its temperature. For water, this energy is required to turn water into steam, and for steam to give off this energy when condensing back to water. At 0.11 MPa, for example, this is around 2201 kJ/kg.

Why is it important?
  • It's important in systems that involve phase changes, like boiling or condensing.
  • Understanding it helps in effectively utilizing steam in heating applications, like in our exercise.
Energy Balance
The concept of energy balance is fundamental in any thermal system. It's about ensuring that the energy going into a system equals the energy going out. This balance is critical when designing systems for heating or cooling. In our exercise, we balance the heat energy given by the steam with the heat absorbed by the water to determine the water's mass flow rate.

Important considerations include:
  • Ensuring no energy is lost in the process, or if it is, quantifying how much.
  • Using energy balance to conserve resources and optimize efficiency.
Saturated Steam
Saturated steam is steam at the point of condensing back into water, sitting on the "edge" between its gaseous and liquid states. It holds maximum energy it can in the steam phase. At 0.11 MPa, the steam is exactly at the temperature where it can start to condense. This type of steam is essential for efficient heat transfer.

How is saturated steam useful?
  • It provides a constant heat output, ideal for steady heating tasks.
  • It ensures consistent and predictable energy releases in industrial applications.
Mass Flow Rate
Mass flow rate measures how much mass of a substance moves across a section over time. It's often expressed in units like kg/s. In heating systems, knowing the mass flow rate of both fluids, like steam and water, is crucial to understanding how much energy is being transferred.

Key reasons this matters include:
  • Ensuring enough heating, cooling, or processing occurs in a given timeframe.
  • Calculating the correct size and type of equipment needed for a processing plant.
Understanding mass flow rate helps in setting operational parameters to meet production goals effectively.

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Most popular questions from this chapter

A counterflow heat exchanger is used to heat \(6.0 \mathrm{~kg} / \mathrm{s}\) of water from \(35^{\circ} \mathrm{C}\) to \(90^{\circ} \mathrm{C}\) by \(14 \mathrm{~kg} / \mathrm{s}\) of oil \(\left(c_{p}=2100 \mathrm{~J} / \mathrm{kg} \mathrm{K}\right)\) supplied at \(150^{\circ} \mathrm{C}\). The design value of the overall heat transfer coefficient is \(120 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). A second unit is to be built at another plant location; however, it has been proposed that the single exchanger be replaced by two smaller counterflow exchangers of equal heat transfer area. The two exchangers are to be connected in series on the water side and in parallel on the oil side. The oil flow is split equally between the two exchangers, and it may be assumed that the overall heat transfer coefficient for the smaller exchangers is also \(120 \mathrm{~W} / \mathrm{m}^{2}\) K. Compare the transfer areas of the oil out two arrangements.

A single-shell-pass, four-tube-pass heat exchanger is tested with water flowing at \(24 \mathrm{~kg} / \mathrm{s}\) in the tubes and oil flowing in the shell at \(20 \mathrm{~kg} / \mathrm{s}\). The tube bundle has 80 steel tubes, \(30 \mathrm{~m}\) long, with a \(24 \mathrm{~mm}\) O.D. and a \(2 \mathrm{~mm}\) wall thickness. In a test, the measured oil inlet and outlet temperatures are \(85.8^{\circ} \mathrm{C}\) and \(48.1^{\circ} \mathrm{C}\) when the water inlet temperature is maintained at \(20^{\circ} \mathrm{C}\). Estimate the outside heat transfer coefficient. Take the oil specific heat as \(2320 \mathrm{~J} / \mathrm{kg} \mathrm{K}\).

An aircraft oil cooler is to be designed to reduce the oil temperature from \(390 \mathrm{~K}\) to \(365 \mathrm{~K}\). The oil (SAE 50 ) flow rate is \(1.5 \mathrm{~kg} / \mathrm{s}\). If the overall heat transfer coefficient is \(140 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) and the entering air temperature is \(310 \mathrm{~K}\), find the necessary transfer area for (i) counterflow. (ii) parallel flow. Assume balanced flow, \(C_{H}=C_{C}\).

A counterflow waste-heat recuperator is designed to achieve an effectiveness of \(0.6\) when heating \(35 \mathrm{~kg} / \mathrm{s}\) of air with \(20 \mathrm{~kg} / \mathrm{s}\) of waste gas. By what factor must the transfer area be increased to raise the effectiveness of \(0.7 ?\) Assume that the overall heat transfer coefficient remains the same, and take \(c_{p}=1000 \mathrm{~J} / \mathrm{kg} \mathrm{K}\) for the air and \(1100 \mathrm{~J} / \mathrm{kg} \mathrm{K}\) for the waste gas.

In a laundry, \(67^{\circ} \mathrm{C}\) dirty wash water is dumped into the drain, and \(70^{\circ} \mathrm{C}\) clean water is required. Presently \(15^{\circ} \mathrm{C}\) water is heated in an electric hot water heater, and the electricity costs 9 cents/ \(\mathrm{kW}\). The water is required at a rate of \(5000 \mathrm{~kg} / \mathrm{h}\), 12 hours per day, 312 days/yr. To conserve energy it is proposed to install a counterflow heat exchanger to preheat the feed to the electric water heater. The installation will cost $$\$ 20,000$$ plus $$\$ 900$$ per square meter of heat exchanger surface. The interest rate to amortize the investment over 12 years is \(10 \%\) per annum. Taxes and insurance are expected to have a fixed cost of \(\$ 500\) per annum plus \(\$ 50 / y r\) per square meter of heat exchanger surface. If the overall heat transfer coefficient is estimated to be \(1000 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\), determine the optimal heat transfer area of the exchanger and the corresponding net annual savings.
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