Question

A 300-ft-long section of a steam pipe whose outer diameter is 4 in passes through an open space at \(50^{\circ} \mathrm{F}\). The average temperature of the outer surface of the pipe is measured to be \(280^{\circ} \mathrm{F}\), and the average heat transfer coefficient on that surface is determined to be \(6 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). Determine \((a)\) the rate of heat loss from the steam pipe and (b) the annual cost of this energy loss if steam is generated in a natural gas furnace having an efficiency of 86 percent, and the price of natural gas is $$\$ 1.10 /$$ therm ( 1 therm \(=100,000\) Btu).

Short answer

Answer: The rate of heat loss from the steam pipe is approximately \(138000\pi \,\text{Btu/h}\), and the annual cost of this energy loss is approximately $47.90 per year.

Step-by-step solution

Calculate the surface area of the pipe

First, we need to calculate the surface area of the steam pipe. The pipe is 300-ft long and has an outer diameter of 4 inches. We will then convert the diameter to feet and can use the formula for the surface area of a cylinder (without the bases) to find the total surface area. The diameter of the pipe in feet is: \(4 \,\text{in} \times \frac{1 \, \text{ft}}{12\,\text{in}} = \frac{1}{3} \,\text{ft}\) So the radius of the pipe is: \(r = \frac{d}{2} = \frac{1}{6} \,\text{ft}\) Now, we can use the formula for the surface area of a cylinder (without bases) which is \( A = 2 \pi r L \), where L is the length of the pipe and r is its radius. \( A = 2 \pi \times \frac{1}{6} \text{ft} \times 300 \text{ft} = 100\pi \,\text{ft}^2\)

Calculate the heat transfer rate

Now, we will use the formula for the heat transfer rate, which is: \(q = h A \Delta T\) where q is the heat transfer rate, h is the heat transfer coefficient, A is the surface area of the pipe, and ΔT is the temperature difference. \(\Delta T = 280^{\circ} \text{F} - 50^{\circ} \text{F} = 230^{\circ} \text{F}\) Now we can find the heat transfer rate: \(q = 6 \frac{\text{Btu}}{\text{h}\cdot\text{ft}^2 \cdot{ }^{\circ}\text{F}} \times 100\pi \,\text{ft}^2 \times 230^{\circ} \text{F} = 138000\pi \,\text{Btu/h}\) (a) Therefore, the rate of heat loss from the steam pipe is \(138000\pi \,\text{Btu/h}\).

Calculate the energy cost per therm

Next, we will calculate the energy cost per therm considering the efficiency of the natural gas furnace. The efficiency of the furnace is 86% which means that for every 100 Btu output, it consumes 100/0.86 = 116.28 Btu of natural gas input. The energy cost per therm can be found by multiplying the price per therm by the consumed energy to generate 100,000 Btu: \(\text{Energy cost per therm} = \frac{\text{Price per therm}}{\text{Furnace efficiency ratio}} = \frac{1.10 \,\text{\$}}{\frac{100,000\,\text{Btu}}{116,280 \, \text{Btu}}} = 1.2635 \,\text{\$}\)

Calculate the annual cost of energy loss

Finally, we will calculate the annual cost of this energy loss considering the given energy cost per therm. To do this, we will first convert the heat loss rate to Btu/year: \(\text{Heat loss rate} = 138000\pi \,\text{Btu/h} \times \frac{8760 \, \text{h}}{\text{year}} \approx 3789325 \, \text{Btu/year}\) Now, we can convert this to the number of therms: \(\text{Number of therms} = 3789325 \,\text{Btu/year} \times \frac{1\,\text{therm}}{100,000\,\text{Btu}} = 37.89325 \text{\,therms/year}\) Now, we can find the annual cost of this energy loss: \(\text{Energy loss cost} = 37.89325\,\text{therms/year} \times 1.2635 \,\text{\$ / therm} \approx 47.90 \,\text{\$ / year}\) (b) Therefore, the annual cost of this energy loss is approximately \(47.90 \,\text{\$ / year}\).

Key concepts

Surface Area of a Pipe

The surface area of a pipe is crucial in determining how much heat is lost as the steam travels through it. This is because the heat transfer from the pipe to the surrounding environment happens all along its surface. To obtain the surface area, we use the formula for the lateral surface area of a cylinder: \( A = 2 \pi r L \) where \( L \) is the length and \( r \) is the radius of the pipe.

Since the radius is half of the diameter, we first convert the diameter to feet for our calculations. In our case, we've converted the 4-inch diameter into feet and halved it to find the radius. We then multiply by the length of the pipe and by \( \pi \) to obtain the total surface area through which heat transfer occurs.

Heat Loss from Steam Pipes

Once we've established the surface area of the steam pipe, we can focus on how heat is lost. The heat transfer rate is calculated using the formula \( q = h A \Delta T \), where \( h \) is the heat transfer coefficient, \( A \) is the surface area, and \(\Delta T \) is the temperature difference between the steam pipe and the surrounding environment.

In our scenario, the steam pipe's outer surface temperature and the ambient temperature are given, allowing us to calculate \( \Delta T \) by subtracting the ambient temperature from the outer surface temperature of the pipe. The product of the heat transfer coefficient, the surface area, and the temperature difference gives us the heat transfer rate or the rate at which heat is lost from the pipe, usually expressed in Btu/h (British thermal units per hour).

Energy Cost Calculation

• Energy Consumption - First, to estimate the cost, we need to calculate the total energy consumed based on the heat loss rate from the steam pipes.
• Furnace Efficiency - Since not all the energy consumed results in heat due to inefficiency, we factor in the furnace efficiency, which tells us how much energy is actually used to produce the steam.
• Cost Per Therm - We then calculate the cost per therm, which is the cost of 100,000 Btu of heat. Since furnaces have an efficiency rate, we adjust the cost per therm by dividing the price per therm by the efficiency ratio.
• Annual Energy Loss Cost - Last, we estimate the annual cost by multiplying the number of therms lost annually due to heat transfer with the adjusted cost per therm.

In essence, the energy cost calculation provides a financial perspective on the efficiency and insulation effectiveness of steam pipes which is critical in industrial settings to manage operational costs effectively.