During a plant visit, it was observed that a \(1.5-\mathrm{m}\)-high and \(1-m\)-wide section of the vertical front section of a natural gas furnace wall was too hot to touch. The temperature measurements on the surface revealed that the average temperature of the exposed hot surface was \(110^{\circ} \mathrm{C}\), while the temperature of the surrounding air was \(25^{\circ} \mathrm{C}\). The surface appeared to be oxidized, and its emissivity can be taken to be \(0.7\). Taking the temperature of the surrounding surfaces to be \(25^{\circ} \mathrm{C}\) also, determine the rate of heat loss from this furnace. The furnace has an efficiency of 79 percent, and the plant pays \(\$ 1.20\) per therm of natural gas. If the plant operates \(10 \mathrm{~h}\) a day, 310 days a year, and thus \(3100 \mathrm{~h}\) a year, determine the annual cost of the heat loss from this vertical hot surface on the front section of the furnace wall.

Step by step solution

01

Calculate heat loss due to radiation

First, let's find the heat loss from the hot surface due to radiation. Using the Stefan-Boltzmann law, the radiative heat transfer (Q_rad) from the surface can be calculated as follows: $$Q_\text{rad} = ε \times σ × A \times (T_\text{hot}^4 - T_\text{surround}^4)$$ Where: - ε: Surface emissivity (0.7) - σ: Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²K⁴) - A: Surface area (1.5m × 1m = 1.5 m²) - T_hot: Temperature of the hot surface in Kelvin (110°C + 273.15 = 383.15 K) - T_surround: Temperature of the surrounding surfaces in Kelvin (25°C + 273.15 = 298.15 K) Now, let's calculate the radiative heat transfer (Q_rad).

02

Calculate heat loss due to convection

Next, we will determine the heat transfer due to convection by using the following correlation for natural convection heat transfer coefficient (h). $$h_{\text{conv}} = 1.31 × (T_\text{hot} - T_\text{air})^{1/3}$$ Where: - T_air: Temperature of the surrounding air in Kelvin (25°C + 273.15 = 298.15 K) Now, we calculate h_conv and use it to find the convective heat transfer (Q_conv) as follows: $$Q_\text{conv} = h_{\text{conv}} × A × (T_\text{hot} - T_\text{air})$$

03

Calculate total heat loss

Now that we have calculated the heat loss due to radiation (Q_rad) and convection (Q_conv), we can find the total heat loss (Q_total) by adding these two values: $$Q_\text{total} = Q_\text{rad} + Q_\text{conv}$$

04

Calculate heat input

To find the annual cost of the heat loss, we first need to calculate the heat input. Since we know the efficiency of the furnace (79%), we can use the following equation: $$Q_\text{input} = \frac{Q_\text{total}}{\text{Efficiency}}$$ Now, let's calculate the heat input (Q_input).

05

Calculate annual cost of heat loss

Now that we have calculated the heat input (Q_input), we can find the annual cost of the heat loss using the cost of natural gas given in the problem. The energy in one therm is equivalent to 100,000 BTU. To convert the heat input to BTU, we can use the following conversion factor: 1 W = 3.412 BTU/h We also know the plant operates 10 h a day, 310 days a year which equals 3100 h a year. Let's calculate the annual heat loss in BTU with the following equation: Annual Heat Loss (BTU) = Q_input × Time × Conversion Factor Now, let's calculate the annual cost of the heat loss using the cost of natural gas: Annual Cost of Heat Loss = (Annual Heat Loss × Cost per therm) / 100,000 Finally, we can find the annual cost of the heat loss from the vertical hot surface on the front section of the furnace wall.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!