Question

A shell-and-tube process heater is to be selected to heat water \(\left(c_{p}=4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) from \(20^{\circ} \mathrm{C}\) to \(90^{\circ} \mathrm{C}\) by steam flowing on the shell side. The heat transfer load of the heater is \(600 \mathrm{~kW}\). If the inner diameter of the tubes is \(1 \mathrm{~cm}\) and the velocity of water is not to exceed \(3 \mathrm{~m} / \mathrm{s}\), determine how many tubes need to be used in the heat exchanger.

Short answer

Question: Determine the number of tubes required for the shell-and-tube process heater. Answer: To calculate the number of tubes required, follow these steps: 1. Calculate the mass flow rate of water using the given heat transfer load, specific heat capacity, and temperature change. 2. Find the flow rate of water in a single tube using the given maximum allowed velocity and inner diameter of the tubes. 3. Calculate the total heat transfer in a single tube using the specific heat capacity, mass flow rate in a single tube, and temperature change. 4. Determine the number of tubes needed by dividing the required heat transfer load by the heat transfer per tube and rounding up to the nearest whole number.

Step-by-step solution

Calculate the mass flow rate of water

To calculate the mass flow rate (\(\dot{m}\)) of water, we will use the formula: \(\dot{m} = \frac{Q}{c_{p} \Delta T}\) where \(Q\) is the heat transfer load, \(c_{p}\) is the specific heat capacity of water, and \(\Delta T\) is the temperature change. Given values: - \(Q = 600 \mathrm{~kW} = 600 \times 10^3 \mathrm{~W}\) - \(c_{p} = 4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) - \(\Delta T = 90^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C} = 70 \mathrm{~K}\) Now, we can calculate the mass flow rate: \(\dot{m} = \frac{600 \times 10^3 \mathrm{~W}}{4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \times 70 \mathrm{~K}}\)

Find the flow rate in a single tube

Given that the maximum allowed velocity of water is \(3 \mathrm{~m} / \mathrm{s}\), we can find the flow rate of water in a single tube (\(\dot{V}_{\mathrm{tube}}\)) using the formula: \(\dot{V}_{\mathrm{tube}} = A_{\mathrm{tube}} \times v_{\mathrm{max}}\) where \(A_{\mathrm{tube}} = \pi d^2/4\) is the cross-sectional area of a single tube, and \(d\) is the inner diameter of the tubes. Given values: - \(d = 1 \mathrm{~cm} = 0.01 \mathrm{~m}\) - \(v_{\mathrm{max}} = 3 \mathrm{~m} / \mathrm{s}\) Now, we can calculate the flow rate in a single tube: \(\dot{V}_{\mathrm{tube}} = \frac{\pi (0.01 \mathrm{~m})^2}{4} \times 3 \mathrm{~m} / \mathrm{s}\)

Calculate the total heat transfer in a single tube

We can find the total heat transfer in a single tube (\(Q_{\mathrm{tube}}\)) using the formula: \(Q_{\mathrm{tube}} = c_{p} \times \dot{m}_{\mathrm{tube}} \times \Delta T\) where \(\dot{m}_{\mathrm{tube}} = \rho \times \dot{V}_{\mathrm{tube}}\) is the mass flow rate of water in a single tube, and \(\rho\) is the density of water. Assuming the density of water is constant at \(1000 \mathrm{~kg} / \mathrm{m}^3\): \(\dot{m}_{\mathrm{tube}}= 1000 \mathrm{~kg} / \mathrm{m}^3 \times \dot{V}_{\mathrm{tube}}\) Now, we can calculate the total heat transfer in a single tube: \(Q_{\mathrm{tube}} = 4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \times \dot{m}_{\mathrm{tube}} \times 70 \mathrm{~K}\)

Determine the number of tubes needed

To find the number of tubes needed (\(n\)), we will divide the required heat transfer load (\(Q\)) by the heat transfer per tube (\(Q_{\mathrm{tube}}\)): \(n = \frac{Q}{Q_{\mathrm{tube}}}\) Now, we can find the number of tubes needed by substituting in the previously calculated values: \(n = \frac{600 \times 10^3 \mathrm{~W}}{Q_{\mathrm{tube}}}\) Since tubes are required to be whole numbers, the number of tubes needed, \(n\), should be rounded up to the nearest whole number.

Key concepts

Shell-and-Tube Heat Exchangers

Shell-and-tube heat exchangers are a type of heat exchanger that is used to transfer heat between two fluids. One fluid flows through a set of tubes while the other fluid flows over the tubes within a larger shell. This design is advantageous because it allows for efficient heat transfer even with different fluid flow velocities or high-pressure differences.
Shell-and-tube heat exchangers are commonly used in industries for processing chemicals, power generation, and HVAC systems. They are favored due to their versatility and ability to handle a variety of temperatures and pressures. The design can also be easily modified by adding or removing tubes to suit different capacity requirements. This makes them an ideal choice for many industrial applications.

Heat Transfer Calculations

In heat exchangers, calculating heat transfer is essential for determining efficiency and sizing requirements. The basic equation for heat transfer is known as the heat energy equation: \[ Q = m \times c_p \times \Delta T \]where:

• \(Q\) is the heat transfer load in watts (W).
• \(m\) is the mass flow rate in kilograms per second (kg/s).
• \(c_p\) is the specific heat capacity in joules per kilogram per degree Kelvin (J/kg·K).
• \(\Delta T\) is the temperature change in Kelvin (K).

By manipulating this formula, engineers can calculate any one variable if the others are known. It is important to select the correct fluid properties at the working temperatures, as they can vary slightly with temperature changes. This calculation means you can ensure that the heat exchanger will function efficiently under the given operating conditions.

Mass Flow Rate Determination

Mass flow rate is critical in determining how a heat exchanger will perform. It represents the amount of mass passing through a given surface per unit time. The mass flow rate of the fluid is calculated using:\[ \dot{m} = \frac{Q}{c_p \times \Delta T} \]An accurate determination of mass flow rate ensures that the heat exchanger can meet the thermal load it is designed for. In the shell-and-tube heat exchanger, the mass flow rate will affect both the rate of heat transfer and the pressure drop. It's important to meet the design specifications to ensure both efficiency and longevity of the equipment.

Specific Heat Capacity

Specific heat capacity is a property of a material that denotes how much heat energy is required to raise the temperature of a given amount of the substance by one degree Kelvin. For water, this value is approximately 4190 J/kg·K. It is a crucial factor in thermal calculations because it directly affects the heat transfer rate.
Understanding specific heat capacity is important because different substances require different amounts of energy to increase their temperature, which impacts the design and efficiency of heat exchangers. In any heating process, knowing the specific heat capacity of the involved fluids helps in designing systems that optimize energy usage.

Fluid Velocity Constraints

Fluid velocity within a heat exchanger needs to be controlled to optimize performance and prevent mechanical issues. High fluid velocities can lead to increased pressure losses and potential damage due to erosion or vibration, while very low velocities can reduce heat transfer efficiency due to insufficient turbulence.
In shell-and-tube heat exchangers, it is vital to maintain a balance. The typical constraint is to keep the maximum fluid velocity under a specified limit, like the 3 m/s constraint for water in the given exercise. Maintaining this balance ensures a stable and efficient heat transfer process and prolongs the lifespan of the heat exchanger by preventing excessive wear and overheating.