Chicken Calorimetry and Scaled Barn Heat Production: 22-Agric-A1 Applied Plant, Animal or Human Physiology, May 2018, Q6

Question 6: Chicken Calorimetry and Scaled Barn Heat Production (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Direct vs. indirect calorimetry. This problem uses a direct calorimeter: the birds are housed in a sealed chamber and their heat output is captured and measured directly from the temperature and humidity rise of a metered air stream passed through the chamber (a ventilated, open-circuit variant of a heat-sink calorimeter) — in contrast to an indirect calorimeter, which infers heat production from the birds' oxygen consumption and carbon dioxide production via a fixed gas-exchange-to-heat conversion (e.g. Brouwer's equation) rather than by capturing the heat itself.

Check: the source lists an "entropy" value at inlet (17.5 kJ/kg) and outlet (43.6 kJ/kg); given the units (kJ/kg) this almost certainly mislabels moist-air specific enthalpy. Neither value is required once dry-bulb temperature, humidity ratio, and the stated specific heat/latent heat constants are given directly, so the calculation below (as instructed by the question) uses only those quantities, not the "entropy" figures.

Given.

Quantity	Value
Airflow rate (at inlet)	1.5×10⁻³ m³/s
Inlet: dry-bulb T / humidity ratio / specific volume	10°C / 0.00294 kg/kg / 0.81 m³/kg
Outlet: dry-bulb T / humidity ratio	24°C / 0.00763 kg/kg
Specific heat of air, $c_p$	1.0 kJ/kg·K
Latent heat of vaporization, $h_{fg}$	2257 kJ/kg
Calorimeter: chicken count / average mass	5 / 2.0 kg
Barn: chicken count / average mass	3000 / 2.5 kg

Find. The specific (W/kg) sensible and latent heat production rates in the calorimeter, and the total sensible heat production of the 3,000-chicken barn.

Approach. Convert the inlet volumetric airflow to a dry-air mass flow rate using the inlet specific volume, then use that mass flow with the sensible-heat ($c_p\,\Delta T$) and latent-heat ($h_{fg}\,\Delta W$) relations across the inlet-to-outlet rise to get the total heat rates, divide by the calorimeter birds' total mass for the specific (W/kg) rates, then rescale the sensible-heat result from the 2.0-kg reference bird to the barn's 2.5-kg bird using the metabolic body-size law (heat production ∝ $M^{0.75}$, not linearly with mass) before multiplying by the barn's 3,000-bird count.

Dry-air mass flow rate. The airflow is measured at the inlet, so divide by the inlet specific volume: $$\dot{m}_{air} = \frac{\dot{V}}{v_{in}} = \frac{1.5\times10^{-3}}{0.81} = \boxed{1.8519\times10^{-3}\ \text{kg/s}}$$
Sensible and latent heat rates in the calorimeter. Applying $c_p\,\Delta T$ and $h_{fg}\,\Delta W$ across the measured inlet-to-outlet rise: $$\dot{Q}_{sens} = \dot{m}_{air}\,c_p\,(T_{out}-T_{in}) = 1.8519\times10^{-3}\times1.0\times(24-10) = \boxed{25.93\ \text{W}}$$ $$\dot{Q}_{lat} = \dot{m}_{air}\,h_{fg}\,(W_{out}-W_{in}) = 1.8519\times10^{-3}\times2257\times(0.00763-0.00294) = \boxed{19.60\ \text{W}}$$
Specific rates per kg of calorimeter bird. The five calorimeter chickens total $5\times2.0 = 10$ kg: $$\dot{q}_{sens} = \frac{25.93}{10} = \boxed{2.593\ \text{W/kg}}, \qquad \dot{q}_{lat} = \frac{19.60}{10} = \boxed{1.960\ \text{W/kg}}$$
Rescale sensible heat from the 2.0-kg reference bird to the barn's 2.5-kg bird. Per-bird sensible heat in the calorimeter is $25.93/5 = 5.185$ W for a 2.0-kg bird. Total metabolic (and hence sensible) heat production scales with body mass to the power 0.75 (Kleiber's law), so: $$\left(\frac{M_{barn}}{M_{cal}}\right)^{0.75} = \left(\frac{2.5}{2.0}\right)^{0.75} = \boxed{1.1822}$$ $$\dot{Q}_{sens,\,barn\ bird} = 5.185 \times 1.1822 = \boxed{6.130\ \text{W per bird}}$$ This is below the value a naive linear (per-kg) scaling would give ($2.593 \times 2.5 = 6.482$ W/bird), which is the trap this two-step approach is designed to avoid.
Total sensible heat production, 3,000-chicken barn. $$\dot{Q}_{sens,\,barn} = 6.130 \times 3000 = \boxed{18{,}390\ \text{W} \approx 18.4\ \text{kW}}$$ This design sensible-heat load (radiative + convective + conductive losses to the barn air and surfaces) is what a ventilation engineer uses to size the barn's winter ventilation rate (or supplemental heat) so as to remove excess sensible heat and hold the barn near its own thermoneutral design temperature.

Quantity	Result
Dry-air mass flow rate	1.852×10⁻³ kg/s
Sensible heat rate, calorimeter	25.93 W (2.593 W/kg)
Latent heat rate, calorimeter	19.60 W (1.960 W/kg)
Mass-scale factor, (2.5/2.0)^0.75	1.1822
Sensible heat per 2.5-kg barn bird	6.130 W
Total sensible heat, 3,000-bird barn	≈ 18.4 kW

Concept & Theory

Topic: Direct-calorimeter sensible/latent heat partitioning from inlet–outlet air-property changes, and metabolic body-size scaling of the result to a different animal mass and flock size.

Key relations: $\dot{Q}_{sens} = \dot{m}_{air}\,c_p\,\Delta T$; $\dot{Q}_{lat} = \dot{m}_{air}\,h_{fg}\,\Delta W$; Kleiber's law: total heat rate $\propto M^{0.75}$, so specific (per-kg) rate $\propto M^{-0.25}$.

Why this works: A ventilated calorimeter is simply a control volume: whatever sensible and latent heat the birds add to the passing air stream must appear as, respectively, a dry-bulb temperature rise and a humidity-ratio rise between inlet and outlet, so each heat rate follows directly from the corresponding air-property change and the (dry-air) mass flow rate. A tabulated specific heat-production rate is only strictly valid at the body mass it was measured on; because heat production scales with mass more slowly than mass itself, scaling to a different reference mass requires the $M^{0.75}$ correction rather than a simple per-kg multiplication.

Common pitfall: Multiplying the 2.593 W/kg specific rate directly by the barn bird's actual 2.5 kg body mass instead of rescaling the calorimeter reference bird's total heat output by $(M_{actual}/M_{ref})^{0.75}$ — this silently assumes linear ($M^{1.0}$) rather than $M^{0.75}$ scaling and overstates the barn's design heat load (by roughly 5-6% here, and more the further the actual mass is from the reference mass); and using the outlet specific volume (0.85 m³/kg) rather than the inlet value to convert the (inlet-measured) volumetric airflow to a mass flow rate.

Source: ASABE Standards, Livestock Energetics and Thermal Environmental Management (direct calorimetry, design sensible heat production); Schmidt-Nielsen, Animal Physiology, 5th ed., Ch. 6 (Metabolic rate and body size).

Question 6 of 6: Chicken Calorimetry and Scaled Barn Heat Production

Question 6: Chicken Calorimetry and Scaled Barn Heat Production (20 marks)