Question

Chickens with an average mass of \(2.2 \mathrm{~kg}\) and average specific heat of \(3.54 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\) are to be cooled by chilled water that enters a continuous-flow-type immersion chiller at \(0.5^{\circ} \mathrm{C}\). Chickens are dropped into the chiller at a uniform temperature of \(15^{\circ} \mathrm{C}\) at a rate of 500 chickens per hour and are cooled to an average temperature of \(3^{\circ} \mathrm{C}\) before they are taken out. The chiller gains heat from the surroundings at a rate of \(210 \mathrm{~kJ} / \mathrm{min}\). Determine \((a)\) the rate of heat removal from the chicken, in \(\mathrm{kW}\), and ( \(b\) ) the mass flow rate of water, in \(\mathrm{kg} / \mathrm{s}\), if the temperature rise of water is not to exceed \(2^{\circ} \mathrm{C}\).

Short answer

Question: Determine (a) the rate of heat removal from the chicken and (b) the mass flow rate of water. Answer: (a) The rate of heat removal from the chicken is approximately \(12.9 \mathrm{kW}\) (b) The mass flow rate of water is approximately \(0.907 \frac{\mathrm{kg}}{\mathrm{s}}\)

Step-by-step solution

(Step 1: Determine the heat balance equation)

The energy balance equation is given by: \(Q_{chicken} + Q_{water} + Q_{surroundings} = 0\) Where \(Q_{chicken}\) is the heat removed from the chicken, \(Q_{water}\) is the heat absorbed by the water, and \(Q_{surroundings}\) is the heat gained from the surroundings.

(Step 2: Express heat gained/removed in terms of mass, specific heat, and temperature change)

The heat gained or removed can be expressed as the product of mass, specific heat, and temperature change. \(Q_{chicken}=\dot{m}_{chicken} \cdot c_{chicken} \cdot \Delta T_{chicken}\) \(Q_{water}=\dot{m}_{water} \cdot c_{water} \cdot \Delta T_{water}\) Where \(\dot{m}\) represents mass flow rate, \(c\) represents specific heat, and \(\Delta T\) represents the temperature change.

(Step 3: Write expressions for temperature change)

The temperature change for chickens is given by: \(\Delta T_{chicken} = 15^{\circ} \mathrm{C} - 3^{\circ} \mathrm{C} = 12^{\circ} \mathrm{C}\) The temperature change for water is given by: \(\Delta T_{water} = 2^{\circ} \mathrm{C}\)

(Step 4: Convert the heat gain from surroundings to \(\mathrm{kW}\))

\(Q_{surroundings} = 210 \frac{\mathrm{kJ}}{\mathrm{min}} \cdot \frac{1 \mathrm{min}}{60\mathrm{s}} = 3.5 \frac{\mathrm{kW}}{\mathrm{s}}\)

(Step 5: Calculate the mass flow rate of chickens by converting the given rate in chickens per hour to kg/s)

\(\dot{m}_{chicken} = 500 \frac{\mathrm{chickens}}{\mathrm{hour}} \cdot \frac{2.2 \mathrm{~kg}}{\mathrm{chicken}} \cdot \frac{1 \mathrm{hour}}{3600\mathrm{sec}} = 0.3056 \frac{\mathrm{kg}}{\mathrm{s}}\)

(Step 6: Calculate the rate of heat removal from the chicken)

Substitute the known values into the energy balance equation and solve for \(Q_{chicken}\): \(Q_{chicken} + Q_{water} + Q_{surroundings} = 0\) \(Q_{chicken} = - Q_{water} - Q_{surroundings}\) \(Q_{chicken} = - \dot{m}_{water} \cdot c_{water} \cdot \Delta T_{water} - 3.5 \mathrm{kW}\) \(Q_{chicken} = - \dot{m}_{water} \cdot c_{water} \cdot 2^{\circ} \mathrm{C} - 3.5 \mathrm{kW}\) Using the known values (\(c_{water}\) is \(4.186 \frac{\mathrm{kJ}}{\mathrm{kg}\cdot^{\circ} \mathrm{C}}\)): \(Q_{chicken}=\dot{m}_{chicken} \cdot c_{chicken}\cdot \Delta T_{chicken}\) \(-3.5\mathrm{kW} + \dot{m}_{water}\cdot 4.186\frac{\mathrm{kJ}}{\mathrm{kg} \cdot\,^{\circ} \mathrm{C}} \cdot 2^{\circ} \mathrm{C} = 0.3056\frac{\mathrm{kg}}{\mathrm{s}}\cdot 3.54\frac{\mathrm{kJ}}{\mathrm{kg} \cdot\,^{\circ} \mathrm{C}}\cdot 12^{\circ} \mathrm{C}\) Now, solve for \(Q_{chicken}\): \(Q_{chicken} \approx 12.9 \mathrm{kW}\)

(Step 7: Calculate the mass flow rate of water)

Rearrange the energy balance equation to solve for \(\dot{m}_{water}\) and plug in the known values: \(\dot{m}_{water} = \frac{- Q_{chicken} + Q_{surroundings}}{c_{water} \cdot \Delta T_{water}}\) \(\dot{m}_{water} = \frac{-12.9 \mathrm{kW} + 3.5 \mathrm{kW}}{4.186\frac{\mathrm{kJ}}{\mathrm{kg} \cdot\,^{\circ} \mathrm{C}} \cdot 2^{\circ} \mathrm{C}}\) \(\dot{m}_{water} \approx 0.907 \frac{\mathrm{kg}}{\mathrm{s}}\)

(Final answers)

(a) The rate of heat removal from the chicken is approximately \(12.9 \mathrm{kW}\) (b) The mass flow rate of water is approximately \(0.907 \frac{\mathrm{kg}}{\mathrm{s}}\)