1. What is an Entering Air Temperature for Duct Calculator?

Definition: This calculator computes the temperature of air entering a duct (\( T_e \)) based on the leaving air temperature, surrounding air temperature, and a dimensionless parameter \( y \), calculated using the air flow rate, duct dimensions, and standard air specific heat (\( C_p = 0.24 \, \text{Btu/lb_f-°F} \)).

Purpose: It is used in HVAC design to determine the inlet air temperature required to achieve a desired outlet temperature, accounting for heat transfer through duct walls.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Entering Air Temperature: \[ T_e = \frac{T_l (y + 1) - 2 T_a}{(y - 1)} \] Dimensionless Constant (Rectangular Ducts): \[ y = \frac{10 A V \rho C_p}{U P L}, \quad A = W \times H, \quad P = 2(W + H), \quad V = \frac{144 Q}{A} \] Dimensionless Constant (Round Ducts): \[ y = \frac{2.5 D V \rho C_p}{U L}, \quad A = \frac{\pi D^2}{4}, \quad V = \frac{144 Q}{A} \] Specific Heat: \( C_p = 0.24 \, \text{Btu/lb_f-°F} \)

Where:

• \( T_e \): Temperature of air entering duct (°F, °C)
• \( T_l \): Temperature of air leaving duct (°F, °C)
• \( T_a \): Temperature of air surrounding duct (°F, °C)
• \( y \): Dimensionless constant
• \( A \): Cross-sectional area of duct (in²)
• \( W \): Width of rectangular duct (in., ft, m)
• \( H \): Height of rectangular duct (in., ft, m)
• \( P \): Perimeter of rectangular duct (in.)
• \( Q \): Air flow rate (cfm, m³/s)
• \( V \): Average velocity (fpm)
• \( \rho \): Air density (lb_f/ft³, kg/m³)
• \( C_p \): Specific heat of air (0.24 Btu/lb_f-°F)
• \( U \): Overall heat transfer coefficient (Btu/hr-ft²-°F, W/m²-K)
• \( L \): Duct length (ft, m)
• \( D \): Diameter of round duct (in., ft, m)

Unit Conversions:

• Temperatures (\( T_l \), \( T_a \), \( T_e \)): °F, °C (°F = °C × 9/5 + 32)
• Heat Transfer Coefficient (\( U \)): Btu/hr-ft²-°F, W/m²-K (1 W/m²-K = 0.176110 Btu/hr-ft²-°F)
• Air Flow Rate (\( Q \)): cfm, m³/s (1 m³/s = 2118.88 cfm)
• Air Density (\( \rho \)): lb_f/ft³, kg/m³ (1 kg/m³ = 0.062428 lb_f/ft³)
• Duct Length (\( L \)): ft, m (1 m = 3.28084 ft)
• Width (\( W \)), Height (\( H \)), Diameter (\( D \)):
  • in.: No conversion
  • ft: 1 ft = 12 in.
  • m: 1 m = 39.3701 in.

Steps:

• Select the duct type (rectangular or round).
• Enter the leaving air temperature (\( T_l \)), surrounding air temperature (\( T_a \)), overall heat transfer coefficient (\( U \)), air flow rate (\( Q \)), air density (\( \rho \)), and duct length (\( L \)).
• For rectangular ducts, enter width (\( W \)) and height (\( H \)); for round ducts, enter diameter (\( D \)).
• Convert all inputs to consistent units (e.g., °F, Btu/hr-ft²-°F, cfm, lb_f/ft³, in., ft).
• Calculate the cross-sectional area (\( A \)) and, for rectangular ducts, perimeter (\( P \)).
• Calculate the average velocity (\( V = \frac{144 Q}{A} \)).
• Calculate the dimensionless constant (\( y \)) based on duct type, using \( C_p = 0.24 \, \text{Btu/lb_f-°F} \).
• Calculate the entering air temperature (\( T_e \)).
• Display \( A \), \( P \) (rectangular only), \( V \), \( y \), and \( T_e \) with 5 decimal places, or in scientific notation if the value is greater than 10,000 or less than 0.00001.

3. Importance of Entering Air Temperature Calculation

Calculating the entering air temperature is crucial for:

• HVAC System Design: Ensures the inlet air temperature is appropriate to achieve the desired outlet temperature, optimizing system performance.
• Energy Efficiency: Helps design ducts to minimize unwanted heat transfer, reducing energy consumption.
• Thermal Comfort: Maintains consistent air temperatures for occupant comfort.

4. Using the Calculator

Examples:

• Example 1: For a rectangular duct with \( T_l = 122 \, \text{°F} \), \( T_a = 40 \, \text{°F} \), \( W = 36 \, \text{in.} \), \( H = 24 \, \text{in.} \), \( Q = 1000 \, \text{cfm} \), \( \rho = 0.075 \, \text{lb_f/ft³} \), \( U = 0.73 \, \text{Btu/hr-ft²-°F} \), \( L = 40 \, \text{ft} \), entering air temperature in °F:
  • \( A = 36 \times 24 = 864 \, \text{in²} \)
  • \( P = 2 \times (36 + 24) = 120 \, \text{in.} \)
  • \( V = \frac{144 \times 1000}{864} \approx 166.67 \, \text{fpm} \)
  • \( y = \frac{10 \times 864 \times 166.67 \times 0.075 \times 0.24}{0.73 \times 120 \times 40} \approx 5.28 \)
  • \( T_e = \frac{122 \times (5.28 + 1) - 2 \times 40}{5.28 - 1} \approx \frac{766.16 - 80}{4.28} \approx 160.32 \)
  • Since all values are < 10000 and > 0.00001, display with 5 decimal places: \( A = 864.00000 \, \text{in²} \), \( P = 120.00000 \, \text{in.} \), \( V = 166.67000 \, \text{fpm} \), \( y = 5.28000 \), \( T_e = 160.32000 \)
• Example 2: For a round duct with \( T_l = 50 \, \text{°C} \), \( T_a = 4.4 \, \text{°C} \), \( D = 0.5 \, \text{m} \), \( Q = 0.5 \, \text{m³/s} \), \( \rho = 1.2 \, \text{kg/m³} \), \( U = 4.1 \, \text{W/m²-K} \), \( L = 12 \, \text{m} \), entering air temperature in °C:
  • Convert: \( T_l = 50 \times 9/5 + 32 = 122 \, \text{°F} \), \( T_a = 4.4 \times 9/5 + 32 = 39.92 \, \text{°F} \)
  • \( D = 0.5 \times 39.3701 \approx 19.68505 \, \text{in.} \)
  • \( Q = 0.5 \times 2118.88 \approx 1059.44 \, \text{cfm} \)
  • \( \rho = 1.2 \times 0.062428 \approx 0.074914 \, \text{lb_f/ft³} \)
  • \( U = 4.1 \times 0.176110 \approx 0.722051 \, \text{Btu/hr-ft²-°F} \)
  • \( L = 12 \times 3.28084 \approx 39.37008 \, \text{ft} \)
  • \( A = \frac{\pi \times 19.68505^2}{4} \approx 304.306 \, \text{in²} \)
  • \( V = \frac{144 \times 1059.44}{304.306} \approx 501.34 \, \text{fpm} \)
  • \( y = \frac{2.5 \times 19.68505 \times 501.34 \times 0.074914 \times 0.24}{0.722051 \times 39.37008} \approx 15.60 \)
  • \( T_e = \frac{122 \times (15.60 + 1) - 2 \times 39.92}{15.60 - 1} \approx \frac{2025.2 - 79.84}{14.60} \approx 133.22 \, \text{°F} \)
  • Convert to °C: \( (133.22 - 32) \times 5/9 \approx 56.23 \)
  • Since all values are < 10000 and > 0.00001, display with 5 decimal places: \( A = 304.30600 \, \text{in²} \), \( V = 501.34000 \, \text{fpm} \), \( y = 15.60000 \), \( T_e = 56.23000 \)
• Example 3: For a rectangular duct with \( T_l = 100 \, \text{°F} \), \( T_a = 80 \, \text{°F} \), \( W = 2 \, \text{ft} \), \( H = 1 \, \text{ft} \), \( Q = 500 \, \text{cfm} \), \( \rho = 0.08 \, \text{lb_f/ft³} \), \( U = 0.5 \, \text{Btu/hr-ft²-°F} \), \( L = 50 \, \text{ft} \), entering air temperature in °F:
  • Convert: \( W = 2 \times 12 = 24 \, \text{in.} \), \( H = 1 \times 12 = 12 \, \text{in.} \)
  • \( A = 24 \times 12 = 288 \, \text{in²} \)
  • \( P = 2 \times (24 + 12) = 72 \, \text{in.} \)
  • \( V = \frac{144 \times 500}{288} = 250 \, \text{fpm} \)
  • \( y = \frac{10 \times 288 \times 250 \times 0.08 \times 0.24}{0.5 \times 72 \times 50} \approx 7.68 \)
  • \( T_e = \frac{100 \times (7.68 + 1) - 2 \times 80}{7.68 - 1} \approx \frac{868 - 160}{6.68} \approx 105.99 \)
  • Since all values are < 10000 and > 0.00001, display with 5 decimal places: \( A = 288.00000 \, \text{in²} \), \( P = 72.00000 \, \text{in.} \), \( V = 250.00000 \, \text{fpm} \), \( y = 7.68000 \), \( T_e = 105.99000 \)

5. Frequently Asked Questions (FAQ)

Q: What does the entering air temperature represent?
A: The entering air temperature (\( T_e \)) is the temperature of air entering a duct, calculated to achieve a specified leaving air temperature, accounting for heat transfer.

Q: How can I determine the input parameters?
A: Leaving and surrounding air temperatures (\( T_l \), \( T_a \)) are measured or estimated. Duct dimensions (\( W \), \( H \), \( D \), \( L \)) are obtained from design specifications. Air flow rate (\( Q \)), density (\( \rho \)), and heat transfer coefficient (\( U \)) are measured or derived from standards. Specific heat is fixed at \( C_p = 0.24 \, \text{Btu/lb_f-°F} \).

Q: Why is the entering air temperature important in HVAC design?
A: It ensures the HVAC system delivers air at the desired temperature, optimizing thermal comfort and energy efficiency.

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