Question

A refrigeration system is being designed to cool \(\operatorname{eggs}\left(\rho=67.4 \mathrm{lbm} / \mathrm{ft}^{3} \text { and } c_{p}=0.80 \mathrm{Btu} / \mathrm{lbm} \cdot^{\circ} \mathrm{F}\right)\) with an average mass of \(0.14 \mathrm{lbm}\) from an initial temperature of \(90^{\circ} \mathrm{F}\) to a final average temperature of \(50^{\circ} \mathrm{F}\) by air at \(34^{\circ} \mathrm{F}\) at a rate of 10,000 eggs per hour. Determine \((a)\) the rate of heat removal from the eggs, in \(\mathrm{Btu} / \mathrm{h}\) and \((b)\) the required volume flow rate of air, in \(\mathrm{ft}^{3} / \mathrm{h}\), if the temperature rise of air is not to exceed \(10^{\circ} \mathrm{F}\).

Short answer

\(\dot{Q}_{eggs}\) = 10,000 x 0.14 x 0.80 x 40 = 44,800 Btu/h #tag_title# (5) Calculate the required volume flow rate of air#tag_content# Using the formula derived in step 3: \(\dot{V} = \frac{N m_e \Delta T}{\rho \Delta T_{air}} = \frac{10,000 \times 0.14 \times 40}{0.075 \times 10} = \frac{56,000}{0.75} = 74,667 ft^3/h\) Therefore, the required volume flow rate of air is 74,667 ft^3/h.

Step-by-step solution

(1) Calculate the total heat removal from the eggs

To find the total amount of heat that must be removed from the eggs, we can use the formula: \(q = mc_p \Delta T\) where \(q\) is the heat transfer, \(m\) is the mass, \(c_p\) is the specific heat, and \(\Delta T\) is the temperature change. The mass is equal to the number of eggs multiplied by the average mass of an egg, and the temperature change is equal to the initial temperature minus the final temperature.

(2) Find the rate of heat removal

To find the rate of heat removal in Btu/h, we need to divide the total heat removal by the time it takes, which is 1 hour. Let's denote \(N\) as the rate of eggs being cooled and \(m_e\) as the mass of an individual egg. Then the rate of heat removal from the eggs is: \(\dot{Q}_{eggs} = N m_e c_p \Delta T\)

(3) Calculate the required volume flow rate of air

The rate of heat removal by air can be expressed as: \(\dot{Q}_{air} = \dot{V} \rho c_p \Delta T_{air}\) where \(\dot{V}\) is the volume flow rate, \(\rho\) is the density of air, and \(\Delta T_{air}\) is the temperature rise of the air which is given to be not more than 10°F. Since the rate of heat removal from the eggs and by the air must be equal: \(\dot{Q}_{eggs} = \dot{Q}_{air}\) Substitute the values for the rates of heat removal from the eggs and by air, we have: \(N m_e c_p \Delta T = \dot{V} \rho c_p \Delta T_{air}\) Now, we can solve for \(\dot{V}\), the required volume flow rate of air: \(\dot{V} = \frac{N m_e \Delta T}{\rho \Delta T_{air}}\) Now, let's plug in the given values and calculate the rate of heat removal from the eggs and the required volume flow rate of air.

(4) Calculate the rate of heat removal from the eggs

Using the given values and using the formula derived in step 2: \(\dot{Q}_{eggs} = N m_e c_p \Delta T = (10,000 \, \text{eggs/h})\times(0.14\, \mathrm{lbm})\times(0.80\, \mathrm{Btu/lbm\cdot^{\circ}F})\times(90^{\circ}F-50^{\circ}F)\)