Question

13-144 In a natural-gas fired boiler, combustion gases pass through 6-m-long 15 -cm-diameter tubes immersed in water at 1 atm pressure. The tube temperature is measured to be \(105^{\circ} \mathrm{C}\), and the emissivity of the inner surfaces of the tubes is estimated to be \(0.9\). Combustion gases enter the tube at 1 atm and \(1200 \mathrm{~K}\) at a mean velocity of \(3 \mathrm{~m} / \mathrm{s}\). The mole fractions of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) in combustion gases are 8 percent and 16 percent, respectively. Assuming fully developed flow and using properties of air for combustion gases, determine \((a)\) the rates of heat transfer by convection and by radiation from the combustion gases to the tube wall and \((b)\) the rate of evaporation of water.

Short answer

1. Find the Nusselt number, Reynolds number, and Prandtl number using correlations for fully developed flow in a tube and properties of air at the average temperature. 2. Calculate the outlet temperature of gases using mass flow rate of the combustion gases and conservation of energy principle. 3. Calculate the rate of heat transfer by convection using the log mean temperature difference (LMTD) method and the heat transfer coefficient. 4. Calculate the rate of heat transfer by radiation using the Stefan-Boltzmann law and the gas film temperature (average of inlet and outlet temperatures). 5. Calculate the rate of evaporation of water using the total heat transfer rate and the specific enthalpy of vaporization.

Step-by-step solution

Find the Nusselt number, Reynolds number, and Prandtl number

We will use the below correlation for a fully developed flow in a tube to find the Nusselt number: Nu = 0.023*Re^0.8*Pr^0.4 To find the Reynolds number, we need the viscosity and properties of air at the inlet temperature: Re = (ρ*u*D)/μ Pr = (cp*μ)/k We will use the properties of air at the average temperature, (1200+373)/2 = 786.5 K. Since the temperature is given in Celsius, we must convert to Kelvin: T_w = 105+273.15=378.15 K. We can find the necessary properties in tables: For air at 786.5 K: ρ = 0.37 kg/m^3, cp = 1087 J/(kg*K), k = 0.040 W/(m*K), μ = 36.8 * 10^{-6} Pa*sec.

Calculate the outlet temperature of gases

To find the outlet temperature of the gases, we will first need to find the mass flow rate of the combustion gases, which can be found using the formula: m_dot = ρ*u*A where A = π*(D^2)/4 is the cross-sectional area of the tube. At a flow rate of 3 m/s, we have: m_dot = 0.37*3*(π*(0.15^2)/4)=0.0524 kg/s We can calculate the heat transfer coefficient (h) using the Nusselt number (Nu) calculated in step 1: h = (Nu*k)/D

Calculate the rate of heat transfer by convection (Q_conv) from the gases to the tube wall using the LMTD method.

We can use the formula for the log mean temperature difference (LMTD) as follows: LMTD = ((T_inlet - T_wall) - (T_outlet - T_wall))/ln((T_inlet - T_wall)/(T_outlet - T_wall)) We will need T_outlet to calculate the LMTD, which can be found by using the conservation of energy principle (ignoring radiation for now): Q_conv=m_dot*cp*(T_inlet-T_outlet) Now we can solve for T_outlet. Once we have the LMTD, we can use the formula for Q_conv: Q_conv = h*A*LMTD

Calculate the rate of heat transfer by radiation (Q_rad) from the gases to the tube wall.

To find the rate of heat transfer by radiation, we can use the Stefan-Boltzmann law: Q_rad = (σ*emissivity*A*(Tg^4 - Tw^4))/((1/emissivity)-1+(A*g/((1/albedo)*A₀))) where σ = 5.67*10^{-8} W/(m²*K^4) is the Stefan-Boltzmann constant, g is the gap between the tubes (0.15 m for these tubes), and albedo is the reflectance of the tube walls, which we can assume to be 1-emissivity. Note that we need to find the gas film temperature (Tg) to calculate Q_rad. For that, we can use the average of the inlet and outlet temperatures of the gas: Tg = (T_inlet+T_outlet)/2

Calculate the rate of evaporation of water (m_evp) using the total heat transfer rate (Q_total).

The total heat transfer rate (Q_total) is the sum of the heat transfer rates by convection and radiation: Q_total = Q_conv + Q_rad Since we know the specific enthalpy of vaporization (h_fg) at 1 atm pressure, we can now determine the rate of evaporation of water (m_evp) using the total heat transfer rate: m_evp = Q_total / h_fg where h_fg = 2257 kJ/kg (using steam tables at 1 atm pressure). And now we have determined both the rates of heat transfer by convection and radiation from the combustion gases to the tube wall, as well as the rate of evaporation of water.

Key concepts

Natural-gas Fired Boiler

A natural-gas fired boiler is an essential component in many heating systems. It uses natural gas as fuel to generate heat. The combustion gases produced during this burning process are made to pass through tubes while submerged in water. This helps in transferring heat from the gases to the water, which then turns to steam.
This process is efficient because natural gas has a high calorific value, meaning it can produce a lot of energy. In our exercise, the gases flow through 6-meter long tubes with a diameter of 15 cm. The gases enter at a high temperature, and the heat is then transferred to the water outside the tubes. This is crucial in converting water into steam, which is often used in power generation or heating applications. Understanding the flow and heat exchange processes in such systems is key to optimizing efficiency.

Nusselt Number

The Nusselt number ( ext{Nu}) is a dimensionless number that is used to characterize heat transfer in fluid flow. It represents the ratio of convective to conductive heat transfer at a boundary in the fluid.
For a fully developed flow in a pipe, like the one described in our original exercise, we use empirical correlations to find the Nusselt number. The most common form is: \[ ext{Nu} = 0.023 imes ext{Re}^{0.8} imes ext{Pr}^{0.4} \] where ext{Re} is the Reynolds number and ext{Pr} is the Prandtl number.

• The Reynolds number ( ext{Re}) helps us understand whether the flow is laminar or turbulent. \
• The Prandtl number ( ext{Pr}) relates the fluid's viscosity to its thermal conductivity. \

In calculating the heat transfer from combustion gases, we consider these numbers at the average temperature in the pipes. These calculations help us determine the efficiency and effectiveness of heat transfer in the boiler system.

Heat Transfer Coefficients

Heat transfer coefficients represent how well heat is transferred between a fluid and a solid surface. The coefficient is crucial in designing systems like boilers because it directly affects the efficiency of heat transfer.
In our exercise context, the heat transfer coefficient ( ext{h}) is needed to calculate the rate of heat transfer by convection. This coefficient can be derived using the Nusselt number as follows: \[ ext{h} = \frac{ ext{Nu} imes ext{k}}{D} \] where ext{Nu} is the Nusselt number, ext{k} is the thermal conductivity of the fluid, and ext{D} is the diameter of the tube.
The heat transfer coefficient is influenced by factors such as:

• Temperature gradient between the fluid and the tube wall. \
• The type and speed of the fluid flow. \
• Physical properties of the fluid, like viscosity and thermal conductivity. \

Accurate determination of this coefficient helps ensure that heat is effectively transferred, improving the boiler performance.

Evaporation Rate Calculation

Calculating the evaporation rate is a critical step to determine how efficiently a boiler converts water into steam. This conversion is directly related to the total heat transfer to the water.
In our original exercise, the evaporation rate, denoted as ext{m_evp}, can be calculated using the sum of heat transferred by convection and radiation, known as ext{Q_total}. The formula used is: \[ ext{m_evp} = \frac{ ext{Q_total}}{h_{fg}} \] where ext{h_{fg}} is the specific enthalpy of vaporization at the given pressure, which is 1 atm here. In practice, knowing the evaporation rate allows engineers to predict how much fuel a boiler consumes and its overall efficiency.
This information helps in optimizing furnace operation and ensuring that boilers deliver the necessary thermal energy efficiently.