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Chapter 8: Problem 53

Heated air required for a food-drying process is generated by passing ambient air at \(20^{\circ} \mathrm{C}\) through long, circular tubes \((D=50 \mathrm{~mm}, L=5 \mathrm{~m})\) housed in a steam condenser. Saturated steam at atmospheric pressure condenses on the outer surface of the tubes, maintaining a uniform surface temperature of \(100^{\circ} \mathrm{C}\). (a) If an airflow rate of \(0.01 \mathrm{~kg} / \mathrm{s}\) is maintained in each tube, determine the air outlet temperature \(T_{m, o}\) and the total heat rate \(q\) for the tube. (b) The air outlet temperature may be controlled by adjusting the tube mass flow rate. Compute and plot \(T_{m \rho}\) as a function of \(\dot{m}\) for \(0.005 \leq \dot{m} \leq\) \(0.050 \mathrm{~kg} / \mathrm{s}\). If a particular drying process requires approximately \(1 \mathrm{~kg} / \mathrm{s}\) of air at \(75^{\circ} \mathrm{C}\), what design and operating conditions should be prescribed for the air heater, subject to the constraint that the tube diameter and length be fixed at \(50 \mathrm{~mm}\) and \(5 \mathrm{~m}\), respectively?

Short Answer

Expert verified
In summary, to determine the air outlet temperature and the total heat transfer rate for the food-drying process, we used the energy balance and logarithmic mean temperature difference equations. We found the air outlet temperature using iterative methods and determined the total heat transfer rate. We computed the air outlet temperature for different mass flow rates in the given range and plotted the results. Lastly, we determined the design and operating conditions for the air heater that satisfy the constraints on tube diameter and length and achieved the desired air outlet temperature and flow rate by iterating over different mass flow rates and tube configurations.

Step by step solution

01

(Step 1: Calculate the outlet temperature of the air)

To find the air outlet temperature, we can use the following energy balance equation: \[(T_{m,o} - T_{m,i}) = UAT_{lm} / C_p \dot{m}\] Where: \(T_{m,o}\) = outlet temperature of the air (掳C) \(T_{m,i}\) = inlet temperature of the air (掳C) \(U\) = overall heat transfer coefficient (W/m虏路K) \(A\) = heat transfer area (m虏) \(T_{lm}\) = logarithmic mean temperature difference (掳C) \(C_p\) = specific heat capacity of air (J/kg路K) \(\dot{m}\) = mass flow rate of air (kg/s) For this problem, \(T_{m,i} = 20 掳C\) and \(\dot{m} = 0.01 \ \text{kg/s}\). We first find the value of \(T_{lm}\) using the equation: \[T_{lm} = \frac{(T_s - T_{m,o})-(T_s - T_{m,i})}{\ln [(T_s - T_{m,o})/(T_s 鈥� T_{m,i})]}\] Where \(T_s = 100 掳C\) is the surface temperature of the tube. Rearrange the equation to find the outlet temperature \(T_{m,o}\) and solve using an iterative method like the Newton-Raphson method.
02

(Step 2: Find the total heat transfer rate)

Once we have the outlet temperature of the air, we can find the total heat transfer rate using the following formula: \[q = C_p \dot{m} (T_{m,o} - T_{m,i})\]
03

(Step 3: Outlet temperature as a function of mass flow rate)

Next, we need to compute the air outlet temperature \(T_{m \rho}\) for different mass flow rates \(\dot{m}\) in the range of 0.005 kg/s to 0.050 kg/s. For each mass flow rate, follow step 1 to find the outlet temperature and plot the results.
04

(Step 4: Determine design and operating conditions for the air heater)

We are given the constraints on the tube diameter and length, and we need to find the design and operating conditions for the air heater to provide approximately 1 kg/s of air at 75 掳C. To accomplish this, we can adjust the mass flow rate of the air and tube properties to achieve the desired air outlet temperature and flow rate. Iterate over different mass flow rates and tube configurations until you find a combination that satisfies the requirements.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Balance
Energy balance is a fundamental concept in heat transfer that involves equating the heat gained or lost by a system to the heat provided to or extracted from it. The principle of energy balance is based on the law of conservation of energy. In the context of our exercise, the heat balance equation is used to determine the outlet temperature of air in a heating process.

The energy balance equation for the air passing through the steam-heated tubes is given by:\[ (T_{m,o} - T_{m,i}) = \frac{UA T_{lm}}{C_p \dot{m}} \]
  • \( U \): Overall heat transfer coefficient, which measures the heat transfer per unit area and temperature difference.
  • \( A \): Surface area over which the heat transfer takes place.
  • \( T_{lm} \): Logarithmic mean temperature difference, a key concept in heat exchanger design.
  • \( C_p \): Specific heat capacity of the air, showing how much energy is required to change the temperature of a unit mass by one degree.
  • \( \dot{m} \): Mass flow rate of the air.
This equation helps in predicting how the incoming air's temperature will change as it absorbs heat from the steam-condensed tube. To solve for the outlet temperature \( T_{m,o} \), often an iterative method is employed due to the integral nature of certain parameters like \( T_{lm} \).
Logarithmic Mean Temperature Difference
The Logarithmic Mean Temperature Difference (LMTD) is crucial when analyzing heat exchangers because it accounts for varying temperature differences across the exchanger's length. It provides a true average of the temperature difference between two flows, ensuring accurate calculations.

The formula for LMTD is:\[ T_{lm} = \frac{(T_s - T_{m,o}) - (T_s - T_{m,i})}{\ln \left(\frac{T_s - T_{m,o}}{T_s - T_{m,i}}\right)} \]In this equation:
  • \(T_s\) denotes the surface temperature of the heat exchange interface, here 100掳C.
  • \(T_{m,o}\) and \(T_{m,i}\) are outlet and inlet temperatures of the air, respectively.

Understanding LMTD helps in recognizing that the temperature difference isn't uniform across the tube; it changes from the inlet to the outlet. LMTD accounts for this change by smoothing out these temperature variations into a singular comparable value. This concept is important for optimizing heat exchanger performance and ensuring the predicted values are closely aligned with actual behavior.
Mass Flow Rate
Mass flow rate refers to the quantity of mass passing through a section per unit of time, usually expressed in kilograms per second (kg/s). It plays a critical role in determining how much energy can be transported in a heat exchanger.

In the given problem, adjusting the mass flow rate directly influences the air's outlet temperature. As mass flow rate affects the dwell time of air inside the heated tubes, it indirectly governs the extent to which the air will absorb heat.
The formula relating mass flow rate and heat transfer rate is:\[ q = C_p \dot{m} (T_{m,o} - T_{m,i}) \]Here:
  • \(q\) is the total heat transfer rate.
  • \(C_p\) is the specific heat capacity of air.
  • \(\dot{m}\) is the mass flow rate of the air.
  • \(T_{m,o}\), \(T_{m,i}\) are the outlet and inlet temperatures of the air, respectively.

Understanding mass flow rate is essential to achieve the desired outlet temperature. Higher flow rates might lead to lower temperatures as there is less time for heat absorption, whereas lower flow rates might result in more heat being absorbed, increasing the outlet temperature. Thus, carefully adjusting the mass flow rate can fine-tune the thermal output to desired specifications.

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Most popular questions from this chapter

Water at \(300 \mathrm{~K}\) and a flow rate of \(5 \mathrm{~kg} / \mathrm{s}\) enters a black, thin-walled tube, which passes through a large furnace whose walls and air are at a temperature of \(700 \mathrm{~K}\). The diameter and length of the tube are \(0.25 \mathrm{~m}\) and \(8 \mathrm{~m}\), respectively. Convection coefficients associated with water flow through the tube and airflow over the tube are \(300 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. (a) Write an expression for the linearized radiation coefficient corresponding to radiation exchange between the outer surface of the pipe and the furnace walls. Explain how to calculate this coefficient if the surface temperature of the tube is represented by the arithmetic mean of its inlet and outlet values. (b) Determine the outlet temperature of the water, \(T_{m, o^{\circ}}\)

A thin-walled tube with a diameter of \(6 \mathrm{~mm}\) and length of \(20 \mathrm{~m}\) is used to carry exhaust gas from a smoke stack to the laboratory in a nearby building for analysis. The gas enters the tube at \(200^{\circ} \mathrm{C}\) and with a mass flow rate of \(0.003 \mathrm{~kg} / \mathrm{s}\). Autumn winds at a temperature of \(15^{\circ} \mathrm{C}\) blow directly across the tube at a velocity of \(5 \mathrm{~m} / \mathrm{s}\). Assume the thermophysical properties of the exhaust gas are those of air. (a) Estimate the average heat transfer coefficient for the exhaust gas flowing inside the tube. (b) Estimate the heat transfer coefficient for the air flowing across the outside of the tube. (c) Estimate the overall heat transfer coefficient \(U\) and the temperature of the exhaust gas when it reaches the laboratory.

The problem of heat losses from a fluid moving through a buried pipeline has received considerable attention. Practical applications include the trans- Alaska pipeline, as well as power plant steam and water distribution lines. Consider a steel pipe of diameter \(D\) that is used to transport oil flowing at a rate \(\dot{m}_{o}\) through a cold region. The pipe is covered with a layer of insulation of thickness \(t\) and thermal conductivity \(k_{i}\) and is buried in soil to a depth \(z\) (distance from the soil surface to the pipe centerline). Each section of pipe is of length \(L\) and extends between pumping stations in which the oil is heated to ensure low viscosity and hence low pump power requirements. The temperature of the oil entering the pipe from a pumping station and the temperature of the ground above the pipe are designated as \(T_{m, i}\) and \(T_{s}\), respectively, and are known. Consider conditions for which the oil (o) properties may be approximated as \(\rho_{o}=900 \mathrm{~kg} / \mathrm{m}^{3}, c_{p, o}=2000\) \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, \quad \nu_{o}=8.5 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}, \quad k_{o}=0.140 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(P r_{o}=10^{4}\); the oil flow rate is \(\dot{m}_{o}=500 \mathrm{~kg} / \mathrm{s}\); and the pipe diameter is \(1.2 \mathrm{~m}\). (a) Expressing your results in terms of \(D, L, z, t, \dot{m}_{o}\), \(T_{m, i}\) and \(T_{s}\), as well as the appropriate oil \((o)\), insulation ( \(i\) ), and soil \((s)\) properties, obtain all the expressions needed to estimate the temperature \(T_{m \rho o}\) of the oil leaving the pipe. (b) If \(T_{s}=-40^{\circ} \mathrm{C}, T_{m, i}=120^{\circ} \mathrm{C}, t=0.15 \mathrm{~m}, k_{i}=0.05\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, k_{s}=0.5 \mathrm{~W} / \mathrm{m}+\mathrm{K}, z=3 \mathrm{~m}\), and \(L=100 \mathrm{~km}\), what is the value of \(T_{m \rho}\) ? What is the total rate of heat transfer \(q\) from a section of the pipeline? (c) The operations manager wants to know the tradeoff between the burial depth of the pipe and insulation thickness on the heat loss from the pipe. Develop a graphical representation of this design information.

A liquid food product is processed in a continuousflow sterilizer. The liquid enters the sterilizer at a temperature and flow rate of \(T_{m, i, h}=20^{\circ} \mathrm{C}, \dot{m}=1 \mathrm{~kg} / \mathrm{s}\), respectively. A time-at-temperature constraint requires that the product be held at a mean temperature of \(T_{m}=90^{\circ} \mathrm{C}\) for \(10 \mathrm{~s}\) to kill bacteria, while a second constraint is that the local product temperature cannot exceed \(T_{\max }=230^{\circ} \mathrm{C}\) in order to preserve a pleasing taste. The sterilizer consists of an upstream, \(L_{k}=5 \mathrm{~m}\) heating section characterized by a uniform heat flux, an intermediate insulated sterilizing section, and a downstream cooling section of length \(L_{c}=10 \mathrm{~m}\). The cooling section is composed of an uninsulated tube exposed to a quiescent environment at \(T_{\infty}=20^{\circ} \mathrm{C}\). The thin-walled tubing is of diameter \(D=40 \mathrm{~mm}\). Food properties are similar to those of liquid water at \(T=330 \mathrm{~K}\). (a) What heat flux is required in the heating section to ensure a maximum mean product temperature of \(T_{m}=90^{\circ} \mathrm{C}\) ? (b) Determine the location and value of the maximum local product temperature. Is the second constraint satisfied? (c) Determine the minimum length of the sterilizing section needed to satisfy the time-at-temperature constraint. (d) Sketch the axial distribution of the mean, surface, and centerline temperatures from the inlet of the heating section to the outlet of the cooling section.

Consider a thin-walled, metallic tube of length \(L=1 \mathrm{~m}\) and inside diameter \(D_{i}=3 \mathrm{~mm}\). Water enters the tube at \(\dot{m}=0.015 \mathrm{~kg} / \mathrm{s}\) and \(T_{m, i}=97^{\circ} \mathrm{C}\). (a) What is the outlet temperature of the water if the tube surface temperature is maintained at \(27^{\circ} \mathrm{C}\) ? (b) If a \(0.5-\mathrm{mm}\)-thick layer of insulation of \(k=0.05\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) is applied to the tube and its outer surface is maintained at \(27^{\circ} \mathrm{C}\), what is the outlet temperature of the water? (c) If the outer surface of the insulation is no longer maintained at \(27^{\circ} \mathrm{C}\) but is allowed to exchange heat by free convection with ambient air at \(27^{\circ} \mathrm{C}\), what is the outlet temperature of the water? The free convection heat transfer coefficient is \(5 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\).
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