/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 106 8.106 Consider the pharmaceutica... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 106

8.106 Consider the pharmaceutical product of Problem 8.27. Prior to finalizing the manufacturing process, test trials are performed to experimentally determine the dependence of the shelf life of the drug as a function of the sterilization temperature. Hence, the sterilization temperature must be carefully controlled in the trials. To promote good mixing of the pharmaceutical and, in turn, relatively uniform outlet temperatures across the exit tube area, experiments are performed using a device that is constructed of two interwoven coiled tubes, each of 10 -mm diameter. The thin-walled tubing is welded to a solid high thermal conductivity rod of diameter \(D_{r}=40 \mathrm{~mm}\). One tube carries the pharmaceutical product at a mean velocity of \(u_{p}=0.1 \mathrm{~m} / \mathrm{s}\) and inlet temperature of \(25^{\circ} \mathrm{C}\), while the second tube carries pressurized liquid water at \(u_{w}=0.12 \mathrm{~m} / \mathrm{s}\) with an inlet temperature of \(127^{\circ} \mathrm{C}\). The tubes do not contact each other but are each welded to the solid metal rod, with each tube making 20 turns around the rod. The exterior of the apparatus is well insulated. (a) Determine the outlet temperature of the pharmaceutical product. Evaluate the liquid water properties at \(380 \mathrm{~K}\). (b) Investigate the sensitivity of the pharmaceutical's outlet temperature to the velocity of the pressurized water over the range \(0.10

Short Answer

Expert verified
In this problem, we analyze the heat exchange between the pharmaceutical product and the pressurized liquid water in a device made of two interwoven coiled tubes. We first establish the energy balance using the heat transfer concept and calculate the outlet temperature of the pharmaceutical product. Then, we evaluate the water properties at 380 K and investigate the sensitivity of the pharmaceutical's outlet temperature to the water's velocity over the range \(0.10 < u_w < 0.25 \: m/s\). Calculating the heat transfer rate for each velocity of water, we can determine the corresponding outlet temperature of the pharmaceutical product and analyze their sensitivity.

Step by step solution

01

Calculate temperature change of the water moving through the tube

To calculate the temperature change of the water moving through the tube, we will establish the energy balance using the heat transfer concept. The heat transfer rate (Q) can be calculated by multiplying the heat transfer coefficient (h), the surface area (A), and the temperature difference (螖T). For the water tube, the surface area can be calculated as follows: \(A_w = 2 \pi r_w L\) Where \(r_w\) is the radius of the tube and \(L\) is the length of the tube. We also need to find the contact area between the pharmaceutical tube and the solid rod, it can be simplified assuming the tube is layed straight: \(A_p = L\) Now, the heat transfer rate for both tubes can be calculated using the following formula: \(Q = h A_w (T_w - T_r) = h A_p (T_r - T_p)\)
02

Simplify the equation and solve for the outlet temperature of the pharmaceutical product T_p

Simplify the equation by dividing both sides by\(h A_p\): \(\frac{T_w - T_r}{T_r - T_p} = \frac{2 \pi r_w L}{L}\) Plug in the given values: \(\frac{127 - T_r}{T_r - 25} = 2 \pi \times (0.005)\) Solve this equation for the temperature of the rod \(T_r\). After finding \(T_r\), we can use this temperature to find the outlet temperature of the pharmaceutical product, \(T_p\).
03

Evaluate the water properties at 380 K

Given that we need to evaluate the liquid water properties at \(380 \:K\), we can refer to standard water property tables. For instance, we can find values for specific heat capacity (c_p), thermal conductivity (k), and dynamic viscosity (渭). These properties will be used in further analysis of the heat transfer between the pharmaceutical product and the pressurized liquid water.
04

Investigate the sensitivity of the pharmaceutical's outlet temperature to the water's velocity

To investigate the sensitivity of the pharmaceutical's outlet temperature to the velocity of the pressurized water, we will vary the velocity of the water (\(u_w\)) over the range of \(0.10 < u_w < 0.25 \: m/s\). We will then calculate the heat transfer rate for each velocity of water, and determine the corresponding outlet temperature of the pharmaceutical product. For each value of \(u_w\), we can apply the same procedure as before to find the outlet temperature of the pharmaceutical product. Finally, we can plot these temperatures against the water velocities to analyze the sensitivity.

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Key Concepts

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

Pharmaceutical Product Manufacturing
Pharmaceutical product manufacturing involves a series of carefully controlled processes to ensure the products are safe and effective. One crucial aspect of this is thermal processing, where heat transfer plays a vital role in achieving the desired properties of a product. Heat exchangers are often utilized in manufacturing to control temperature during processes like sterilization and product mixing.

Proper temperature control is essential to maintain the integrity and efficacy of pharmaceutical products. This involves the precise manipulation of heat transfer within devices that are often complex, incorporating various materials and designs suited to specific pharmaceutical applications.
  • Utilization of coiled tubes and metal rods can aid in achieving uniform heating, essential for consistent product quality.
  • Heat exchangers can be designed to cater specifically to the viscoelastic and thermal conductivity properties of the pharmaceutical materials.
Given these complexities, meticulous experimental testing is imperative to optimize the manufacturing process and achieve desired product outcomes.
Sterilization Temperature Dependence
Sterilization is a critical process in pharmaceutical manufacturing that significantly affects the shelf life and efficacy of products. The dependence of sterilization on temperature requires that this parameter be carefully controlled and monitored.

The response of pharmaceutical products to changes in sterilization temperature can be studied through experiments that adjust the temperature and observe its effects on the shelf life and stability of the product. Temperature variability can affect the physical and chemical properties of the pharmaceuticals, potentially leading to degradation or loss of potency.
  • Higher temperatures generally increase the rate of sterilization but can compromise the product's stability if not managed correctly.
  • Conversely, lower temperatures may extend stability but could reduce the efficacy of sterilization.
Balancing these aspects is crucial in designing processes that maximize product longevity without compromising safety and effectiveness.
Heat Exchanger Design
In the context of pharmaceutical manufacturing, the design of heat exchangers is critical to ensure the controlled transfer of heat during processes like sterilization and product heating. The choice of materials and configuration directly impacts the performance of a heat exchanger.

A typical heat exchanger design, like the one described in the exercise, consists of two interwoven coiled tubes that transport different fluids. These tubes might be attached to a solid rod to improve thermal efficiency by facilitating heat conduction.
  • The dimensions of the tubes, including their diameter and length, are key parameters influencing the surface area available for heat exchange.
  • The materials used, often metals with high thermal conductivity, ensure efficient heat transfer between the two fluids.
  • The velocity of the fluids running through the tubes is a critical factor; changing the velocity can alter the rate of heat transfer.
By manipulating these variables, engineers can design heat exchangers that optimize temperature control, enhance energy efficiency, and maintain quality standards in pharmaceutical products.

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Most popular questions from this chapter

8.105 The mold used in an injection molding process consists of a top half and a bottom half. Each half is \(60 \mathrm{~mm} \times 60 \mathrm{~mm} \times 20 \mathrm{~mm}\) and is constructed of metal \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=450 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\right)\). The cold mold \(\left(100^{\circ} \mathrm{C}\right)\) is to be heated to \(200^{\circ} \mathrm{C}\) with pressurized water (available at \(275^{\circ} \mathrm{C}\) and a total flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) ) prior to injecting the thermoplastic material. The injection takes only a fraction of a second, and the hot mold \(\left(200^{\circ} \mathrm{C}\right)\) is subsequently cooled with cold water (available at \(25^{\circ} \mathrm{C}\) and a total flow rate of \(0.02 \mathrm{~kg} / \mathrm{s})\) prior to ejecting the molded part. After part ejection, which also takes a fraction of a second, the process is repeated. (a) In conventional mold design, straight cooling (heating) passages are bored through the mold in a location where the passages will not interfere with the molded part. Determine the initial heating rate and the initial cooling rate of the mold when five 5 -mm-diameter, 60-mm-long passages are bored in each half of the mold (10 passages total). The velocity distribution of the water is fully developed at the entrance of each passage in the hot (or cold) mold. (b) New additive manufacturing processes, known as selective freeform fabrication, or \(S F F\), are used to construct molds that are configured with conformal cooling passages. Consider the same mold as before, but now a 5 -mm- diameter, coiled, conformal cooling passage is designed within each half of the SFF-manufactured mold. Each of the two coiled passages has \(N=2\) turns. The coiled passage does not interfere with the molded part. The conformal channels have a coil diameter \(C=50 \mathrm{~mm}\). The total water flow remains the same as in part (a) \((0.01 \mathrm{~kg} / \mathrm{s}\) per coil). Determine the initial heating rate and the initial cooling rate of the mold. (c) Compare the surface areas of the conventional and conformal cooling passages. Compare the rate at which the mold temperature changes for molds configured with the conventional and conformal heating and cooling passages. Which cooling passage, conventional or conformal, will enable production of more parts per day? Neglect the presence of the thermoplastic material.

A novel scheme for dissipating heat from the chips of a multichip array involves machining coolant channels in the ceramic substrate to which the chips are attached. The square chips \(\left(L_{c}=5 \mathrm{~mm}\right)\) are aligned above each of the channels, with longitudinal and transverse pitches of \(S_{L}=S_{T}=20 \mathrm{~mm}\). Water flows through the square cross section \((W=5 \mathrm{~mm}\) ) of each channel with a mean velocity of \(u_{m}=1 \mathrm{~m} / \mathrm{s}\), and its properties may be approximated as \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}\), \(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=855 \times 10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, k=0.610\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\), and \(\operatorname{Pr}=5.8\). Symmetry in the transverse direction dictates the existence of equivalent conditions for each substrate section of length \(L_{s}\) and width \(S_{T}\). (a) Consider a substrate whose length in the flow direction is \(L_{s}=200 \mathrm{~mm}\), thereby providing a total of \(N_{L}=10\) chips attached in-line above each flow channel. To a good approximation, all the heat dissipated by the chips above a channel may be assumed to be transferred to the water flowing through the channel. If each chip dissipates \(5 \mathrm{~W}\), what is the temperature rise of the water passing through the channel? (b) The chip-substrate contact resistance is \(R_{\mathrm{t}, c}^{\mathrm{r}}=\) \(0.5 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), and the three-dimensional conduction resistance for the \(L_{s} \times S_{T}\) substrate section is \(R_{\text {cond }}=0.120 \mathrm{~K} / \mathrm{W}\). If water enters the substrate at \(25^{\circ} \mathrm{C}\) and is in fully developed flow, estimate the temperature \(T_{c}\) of the chips and the temperature \(T_{s}\) of the substrate channel surface.

Many of the solid surfaces for which values of the thermal and momentum accommodation coefficients have been measured are quite different from those used in micro- and nanodevices. Plot the Nusselt number \(N u_{D}\) associated with fully developed laminar flow with constant surface heat flux versus tube diameter for \(1 \mu \mathrm{m} \leq D \leq 1 \mathrm{~mm}\) and (i) \(\alpha_{s}=1, \alpha_{p}=1\), (ii) \(\alpha_{t}=0.1, \alpha_{p}=0.1\), (iii) \(\alpha_{t}=1, \alpha_{p}=0.1\), and (iv) \(\alpha_{t}=0.1, \alpha_{p}=1\). For tubes of what diameter do the accommodation coefficients begin to influence convection heat transfer? For which combination of \(\alpha_{t}\) and \(\alpha_{p}\) does the Nusselt number exhibit the least sensitivity to changes in the diameter of the tube? Which combination results in Nusselt numbers greater than the conventional fully developed laminar value for constant heat flux conditions, \(N u_{D}=4.36\) ? Which combination is associated with the smallest Nusselt numbers? What can you say about the ability to predict convection heat transfer coefficients in a small-scale device if the accommodation coefficients are not known for material from which the device is fabricated? Use properties of air at atmospheric pressure and \(T=300 \mathrm{~K}\).

An experiment is designed to study microscale forced convection. Water at \(T_{\text {mi }}=300 \mathrm{~K}\) is to be heated in a straight, circular glass tube with a \(50-\mu \mathrm{m}\) inner diameter and a wall thickness of \(1 \mathrm{~mm}\). Warm water at \(T_{\infty}=350 \mathrm{~K}, V=2 \mathrm{~m} / \mathrm{s}\) is in cross flow over the exterior tube surface. The experiment is to be designed to cover the operating range \(1 \leq R e_{D} \leq 2000\), where \(R e_{D}\) is the Reynolds number associated with the internal flow. (a) Determine the tube length \(L\) that meets a design requirement that the tube be twice as long as the thermal entrance length associated with the highest Reynolds number of interest. Evaluate water properties at \(305 \mathrm{~K}\). (b) Determine the water outlet temperature, \(T_{\text {mo }}\) that is expected to be associated with \(R e_{D}=2000\). Evaluate the heating water (water in cross flow over the tube) properties at \(330 \mathrm{~K}\). (c) Calculate the pressure drop from the entrance to the exit of the tube for \(R e_{D}=2000\). (d) Based on the calculated flow rate and pressure drop in the tube, estimate the height of a column of water (at \(300 \mathrm{~K}\) ) needed to supply the necessary pressure at the tube entrance and the time needed to collect \(0.1\) liter of water. Discuss how the outlet temperature of the water flowing from the tube, \(T_{m, o}\), might be measured.

An extremely effective method of cooling high-powerdensity silicon chips involves etching microchannels in the back (noncircuit) surface of the chip. The channels are covered with a silicon cap, and cooling is maintained by passing water through the channels. Consider a chip that is \(10 \mathrm{~mm} \times 10 \mathrm{~mm}\) on a side and in which fifty 10 -mm-long rectangular microchannels, each of width \(W=50 \mu \mathrm{m}\) and height \(H=200 \mu \mathrm{m}\), have been etched. Consider operating conditions for which water enters each microchannel at a temperature of \(290 \mathrm{~K}\) and a flow rate of \(10^{-4} \mathrm{~kg} / \mathrm{s}\), while the chip and cap are at a uniform temperature of \(350 \mathrm{~K}\). Assuming fully developed flow in the channel and that all the heat dissipated by the circuits is transferred to the water, determine the water outlet temperature and the chip power dissipation. Water properties may be evaluated at \(300 \mathrm{~K}\).
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