91Ó°ÊÓ

An air duct heater consists of an aligned array of electrical heating elements in which the longitudinal and transverse pitches are \(S_{L}=S_{T}=24 \mathrm{~mm}\). There are 3 rows of elements in the flow direction \(\left(N_{L}=3\right)\) and 4 elements per row \(\left(N_{T}=4\right)\). Atmospheric air with an upstream velocity of \(12 \mathrm{~m} / \mathrm{s}\) and a temperature of \(25^{\circ} \mathrm{C}\) moves in cross flow over the elements, which have a diameter of \(12 \mathrm{~mm}\), a length of \(250 \mathrm{~mm}\), and are maintained at a surface temperature of \(350^{\circ} \mathrm{C}\). (a) Determine the total heat transfer to the air and the temperature of the air leaving the duct heater. (b) Determine the pressure drop across the element bank and the fan power requirement. (c) Compare the average convection coefficient obtained in your analysis with the value for an isolated (single) element. Explain the difference between the results. (d) What effect would increasing the longitudinal and transverse pitches to \(30 \mathrm{~mm}\) have on the exit temperature of the air, the total heat rate, and the pressure drop?

Step by step solution

01

a) Total Heat Transfer and Exit Air Temperature

We'll use the following steps to determine the total heat transfer and the temperature of the air leaving the duct heater. 1. Calculate the Reynolds number for the flow over the elements, using the given air velocity and diameter. Use the air properties at the film temperature, which is the average of the air and surface temperatures: \[ Re = \frac{Ud}{\nu} \] 2. Calculate the Nusselt number using an appropriate correlation, such as the Zukauskas correlation for crossflow over tubes: \[ Nu = CRe^mPr^n \left(\frac{Pr}{Pr_s}\right)^{0.25} \] 3. Calculate the convection heat transfer coefficient, h, using the Nusselt number and thermal conductivity of the air: \[ h = \frac{kNu}{d} \] 4. Calculate the total heat transfer from the element array, Q, by multiplying the convection heat transfer coefficient by the total heat transfer area and the temperature difference between the surface and the air: \[ Q = hA_s(T_s - T_a) \] 5. Determine the exit air temperature using the heat transfer calculated and an appropriate heat transfer relation: \[ T_{a,exit} = T_{a} + \frac{Q}{\dot{m}c_p} \]

02

b) Pressure Drop and Fan Power Requirement

We'll use the following steps to determine the pressure drop across the element bank and the fan power requirement. 1. Calculate the resistance factor, K, for the flow through the element bank. For a regular element array, use a suitable correlation like the Idelchick correlation: \[ K = K_1 + K_2\frac{1}{N_L}\] 2. Calculate the pressure drop across the element bank using Bernoulli's equation and the resistance factor calculated in step 1: \[ \Delta P = \frac{1}{2}\rho U^2 K \] 3. Compute the fan power requirement, P_fan, by multiplying the pressure drop by the volumetric flow rate and dividing it by the fan efficiency: \[ P_{fan} = \frac{\Delta P \dot{V}}{\eta_{fan}} \]

03

c) Comparing Convection Coefficients

To compare the average convection coefficients, calculate the coefficient for an isolated (single) element using the same process as in part a) but considering only one element. Explain the difference in the results based on the influence of neighboring elements in the array and possible changes in flow patterns.

04

d) Effect of Increasing Pitches on Exit Temperature, Heat Rate, and Pressure Drop

To determine the effect of increasing pitches on exit temperature, heat rate, and pressure drop, repeat parts a) and b) using the new longitudinal and transverse pitches of 30 mm. Compare the results with the original configuration and discuss the changes in heat transfer and pressure drop based on the increased spacing and flow characteristics.

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Key Concepts

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

Reynolds Number Calculation

The Reynolds number is a crucial element in understanding how fluid flows over objects. It helps us determine the nature of the flow, whether it's laminar or turbulent. In the case of ducts and tubes, the Reynolds number, which is dimensionless, is calculated using the formula:\[ Re = \frac{Ud}{u} \]Here, \(U\) is the velocity of the fluid, \(d\) is the characteristic length (or diameter of the tubes), and \(u\) is the kinematic viscosity of the fluid. Kinematic viscosity is different for each fluid and varies with temperature, so it's important to use the average (or film) temperature when calculating this number. This temperature is the average of the air and the surface temperature, which gives a more accurate depiction of the fluid behavior.

• Reynolds number below 2300 indicates laminar flow.
• Reynolds number above 4000 indicates turbulent flow.

Understanding whether the flow is laminar or turbulent helps us predict how effectively the heat will be transferred to or from the flowing fluid.

Nusselt Number Correlations

The Nusselt number is a dimensionless number that relates to convective heat transfer. It allows us to find the convective heat transfer coefficient, which is essential for calculating the amount of heat transferred from the surface to the fluid. Calculating the Nusselt number accurately is fundamental to effective heat exchanger design. Zukauskas, among other correlations, is commonly used for calculating the Nusselt number in crossflow over tubes:\[ Nu = CRe^mPr^n \left(\frac{Pr}{Pr_s}\right)^{0.25} \]In this equation, \(Re\) is the Reynolds number, \(Pr\) is the Prandtl number of the fluid, and \(Pr_s\) is the Prandtl number at surface temperature. The constants \(C\), \(m\), and \(n\) are determined from empirical correlations based on the configuration of the array.

• The Prandtl number, \(Pr\), is a measure of the fluid's thermal diffusivity, impacting heat transfer rate.
• Choosing the correct correlation is vital. It depends on the geometry and alignment of the heat exchanger elements.

Pressure Drop in Heat Exchange Systems

Pressure drops within heat exchangers can significantly affect their efficiency and the power required to maintain flow through them. Calculating the pressure drop is crucial for ensuring the system operates efficiently. Let's break it down:The resistance factor \(K\) for an element array is determined using correlations like Idelchick's:\[ K = K_1 + K_2\frac{1}{N_L} \]The pressure drop \(\Delta P\) across the element bank is then calculated with:\[ \Delta P = \frac{1}{2}\rho U^2 K \]This formula incorporates the density \(\rho\) and velocity \(U\) of the fluid. The pressure drop influences the fan power requirements since overcoming this resistance is crucial for sustaining the fluid flow.

• High pressure drop means more energy is needed to move the air through the duct.
• Design considerations should aim to minimize pressure drop for cost-effectiveness.

Convection Heat Transfer

Convection is the heat transfer method that occurs through the movement of fluids. In the context of heat exchangers, convection transfers heat from a hot surface to cooler air moving past it. The effectiveness of this process depends on the convection heat transfer coefficient \(h\), which is found from the Nusselt number:\[ h = \frac{kNu}{d} \]Where \(k\) is the thermal conductivity of the fluid and \(d\) is the characteristic length, such as the diameter of the tubes.
Key factors influencing convection in ducts include:

• Surface area: More surface exposure leads to better heat transfer.
• Temperature difference: Greater differences enhance heat transfer.
• Fluid velocity: Higher speeds generally improve convection efficiency.

Changes in surface roughness, temperature gradients, and flow velocity all impact the rate of heat transfer by convection. The design of the duct system needs to consider these factors to optimize the heat transfer process.