112 年 國立清華大學化學工程學系碩士班《單元操作與輸送現象》

For the description of heat and mass transfer in the fish during the cooking process, which statement is NOT true?

(A) Internal mass transfer is a pure diffusion process described by Fick's law, and the pressure-driven transport of moisture in the muscle can be neglected.

(B) Heat is transferred from the hot air to the surface of the fish by convection and radiation, and from the baking plate by conduction.

(C) Mass transport through the fish is only driven by diffusion and convection.

(D) Fourier's law of conduction can be used to model heat transfer in fish.

(E) Convective boundary conditions are given by Newton's law of cooling.

The governing equation of heat transfer in the fish can be described by the following equation:

$\rho c_p\left(\frac{\partial T}{\partial t}\right)=k\nabla^2T-\rho_w c_{p,w}\mathbf{v}_w\cdot\nabla T$

Here, $\rho$, $c_p$, and $k$ are the density, specific heat, and thermal conductivity of fish, respectively. $\rho_w$, $c_{p,w}$, and $\mathbf{v}_w$ are the density, specific heat, and flow velocity of water transported within the fish, respectively. Which statement is NOT true?

(A) $k\nabla^2T$ is the heat conduction term.

(B) Thermal conductivity can vary with sample geometry.

(C) Radiative transfer is not considered in the governing equation.

(D) Viscous dissipation is ignored in the system.

(E) $\nabla$ is the three-dimensional del operator.

Please choose the correct temperature profile at the geometric center of the fish during the cooking process. $t$ is the cooking time and $T$ is the temperature.

🖼️【此處有附圖，請對照原卷】
（附圖：五張 $T$（°C）對 $t$（min）曲線。(A) 直線上升；(B) 上凹、越升越快；(C) 下凹、先快後慢地上升；(D) 起初急升，很快轉平成平台；(E) S 形：先平緩、中段上升、最後趨平。）

About the heat transfer in the fish, which statement is NOT true?

(A) The unit of heat transfer coefficient is $\mathrm{W/(m^2\cdot K)}$.

(B) Thermal conductivity of fish increases with temperature.

(C) Biot number is the ratio of internal conductive resistance within the body to the external convective resistance at the surface of the body.

(D) Brinkman number in this case is very large.

(E) The unit of thermal diffusivity is the same as the mass diffusivity.

Heat transfer due to radiation should be considered during the cooking process. Please estimate the heat generation due to radiation. Assume the surface temperature of the fish is $100^\circ\mathrm{C}$, the oven temperature is $140^\circ\mathrm{C}$, the emissivity of the oven is 0.9, the surface area of the fish is $48\ \mathrm{cm^2}$, and the Stefan–Boltzmann constant is $5.676\times10^{-8}\ \mathrm{W/(m^2\cdot K^4)}$.

(A) 0.07 W

(B) 2.4 W

(C) 48 W

(D) 1728 W

(E) 23875 W

The governing equation of mass transfer in the fish is based on the following equation:

$\frac{\partial c}{\partial t}=D_w\nabla^2c-\nabla\cdot(\mathbf{v}_w c)$

Here, $c$ and $D_w$ are the moisture (water) concentration and moisture diffusion coefficient in the fish, respectively. Which statement is NOT true?

(A) Water density is constant in this case.

(B) Diffusion coefficient does not change with location.

(C) The system only has free convection of mass transfer.

(D) The system is able to reach a steady state.

(E) $\nabla\cdot\mathbf{v}_w c$ can be written as $\mathbf{v}_w\cdot\nabla c$ when the density of water and diffusion coefficient are constant.

The flow velocity of the water in the fish is derived from the following equation:

$\mathbf{v}_w=-\frac{\kappa G'}{\mu_w}\nabla\left(w_w-w_{w,\mathrm{eq}}(T)\right)$

Here, $\kappa$ is the permeability of fish, $G'$ is the storage modulus of the fish, $\mu_w$ is the dynamic viscosity of water, $w_w$ is the mass fraction of water in the fish, and $w_{w,\mathrm{eq}}$ is the water holding capacity of the fish as a function of temperature $T$. Which statement is NOT true?

(A) The equation is derived from Darcy's law when the porous medium approach is applied.

(B) The driving force of the equation is the difference in temperature.

(C) Water holding capacity decreases when the temperature increases.

(D) Storage modulus increases when the temperature increases.

(E) Viscosity decreases when the temperature increases.

Please choose the correct mass fraction profile of fish during the cooking process. $t$ is the cooking time and $w$ is the mass fraction of water in the fish.

$w=\frac{cM_w}{\rho_{\mathrm{fish}}},\qquad M_w\text{ is the molecular weight of water.}$

🖼️【此處有附圖，請對照原卷】
（附圖：五張 $w$（kg/kg）對 $t$（min）曲線。(A) 直線下降；(B) 下凹、先緩後陡地下降；(C) 上凹、先陡後緩地下降並趨近定值；(D) 起初驟降，隨即轉平；(E) 反 S 形：先平緩、中段下降、最後趨平。）

About the mass transfer in the fish, which statement is NOT true?

(A) The unit of mass diffusivity is $\mathrm{m^2/s}$.

(B) Mass diffusivity of water increases with temperature.

(C) Sherwood number is a dimensionless mass transfer coefficient that represents the ratio of the convective mass transfer rate to the mass diffusion rate.

(D) Péclet number can describe the ratio of momentum diffusivity and mass diffusivity.

(E) The unit of thermal diffusivity is the same as the mass diffusivity.

To obtain the weight loss of the fish, it is critical to obtain the evaporation rate. Please calculate the evaporation rate ($w_{A0}$) during the cooking process at isothermal conditions. Assume that the water mass fraction is 0.8, the surface area of the fish is $48\ \mathrm{cm^2}$, the mass transfer coefficient is $0.001\ \mathrm{m/s}$, the relative humidity is 0%, and water does not transport back to the fish.

Hint: $w_{A0}-w_w(w_{A0}+w_{B0})=k_wA\Delta w_w$.

(A) $0.192\ \mathrm{m^3/s}$

(B) $0.294\ \mathrm{m^3/s}$

(C) $0.54\ \mathrm{m^3/s}$

(D) $1.024\ \mathrm{m^3/s}$

(E) $1.42\ \mathrm{m^3/s}$

You probably would like to draw a qualitative picture of the diffusion of diluted gases first. From a molecular point of view, how does diffusion actually occur for gas molecules?

(A) The diffusion of gas molecules is driven by a pressure gradient. They flow from places of high pressure to regions of low pressure.

(B) The diffusion of gas molecules is driven by the van der Waals forces between individual molecules. The collective attraction forces result in a net movement of the molecules.

(C) The diffusion of gas molecules is driven by the random collision of molecules. The collective collision events result in a net movement of the molecules.

(D) The diffusion of gas molecules is driven by electrostatic interactions between the molecules. The attractive and repulsive interactions between the charged rubbing gas molecules result in a net movement of the molecules.

Now, you start to think about how to describe the diffusion of gas molecules mathematically. To do so, you need some properties of the gas molecules and must consider which properties directly affect diffusion behavior and therefore are needed to form your mathematical formula.

(A) Number density ($n$), diameter ($d$), charge ($z$).

(B) Heat capacity ($C_p$), diameter ($d$), mass ($M$).

(C) Mass ($M$), diameter ($d$).

(D) Number density ($n$), mass ($M$).

(E) Number density ($n$), diameter ($d$), mass ($M$).

You also need some system parameters or constants to complete your description. From the list below, choose those that are required.

(A) Temperature, Reynolds number, volume.

(B) Pressure, Boltzmann constant, volume.

(C) Boltzmann constant, temperature.

(D) Reynolds number, temperature.

(E) Boltzmann constant, Reynolds number, temperature.

The number of gas molecules passing through a unit surface, i.e., the diffusional flux, can be modeled by a mass balance across a particular plane at $y$ with the simplified assumption that the gas molecules, on average, will meet another molecule after traveling a distance $\lambda$ in space with a frequency of $n\bar{u}$, where $\bar{u}$ is the mean molecular speed of the gas molecules.

If there is a mass fraction ($w_a$) gradient of molecules of gas $a$, this means that the diffusional flux of gas molecules across the plane at $y$ can be expressed by which of the following equations, which can be equated to Fick's first law of diffusion to solve for the diffusion coefficient?

🖼️【此處有附圖，請對照原卷】

(A) $j_a=-2n\bar{u}\lambda\frac{dw_a}{dy}$

(B) $j_a=-4n\bar{u}M\frac{dw_a}{dy}$

(C) $j_a=-2n\bar{u}M\lambda\frac{dw_a}{dy}$

(D) $j_a=-2\rho\bar{u}M\lambda\frac{dw_a}{dy}$

(E) $j_a=-4\rho\bar{u}M\lambda\frac{dw_a}{dy}$

The diffusion coefficient is therefore which of the following?

(A) $D_{aa}=2n\bar{u}\lambda$

(B) $D_{aa}=4\bar{u}M\lambda$

(C) $D_{aa}=2n\bar{u}M\lambda$

(D) $D_{aa}=2\bar{u}\lambda$

(E) $D_{aa}=4n\bar{u}\lambda$

Now you know: by how many times does hydrogen diffuse faster than tritium?

(A) 1.414

(B) 1.732

(C) 2.000

(D) 3.000

(E) 4.000

Which of the following graphs best describes the evaporation rate over time for the hydrated (supersaturated) porous medium?

🖼️【此處有附圖，請對照原卷】
（附圖：五張蒸發速率 $R$（g/min）對 $t$（min）曲線。(A) 先持平，短暫跳升到較高的平台，之後下降成一段緩坡，再陡降並以曲線趨近零；(B) 先上升到平台，接著隆起一個峰後下降至零；(C) 高平台後直線下降到較低的平台，再直線下降至零；(D) 從低處緩升成鐘形峰，再下降至零；(E) 先直線上升到平台，平台後先以較緩的斜率下降，再以曲線下降至零。）

During the constant rate period, the system is essentially identical to which of the following?

(A) Evaporation of water droplets on a $200^\circ\mathrm{C}$ pan.

(B) Evaporation of a puddle of muddy water after a rainy day.

(C) Evaporation of hot water placed inside a cup.

(D) Drying of a wet mop with the excess water already squeezed out.

(E) Evaporation of water droplets sprinkled from a nozzle into hot air.

What is the rate of evaporation in the constant rate period if you are heating your hydrated material from $25^\circ\mathrm{C}$? The input power is $1000\ \mathrm{W}$. The specific heat capacity of the solid porous material is $2\ \mathrm{J/(g\cdot^\circ C)}$. The specific heat capacity of liquid water is $4.18\ \mathrm{J/(g\cdot^\circ C)}$, and that of water vapor is $1.9\ \mathrm{J/(g\cdot^\circ C)}$. The latent heat of water evaporation is $2.256\ \mathrm{kJ/g}$. The dry solid is $500\ \mathrm{g}$ and the wet solid is initially $1000\ \mathrm{g}$.

(A) $1.36\ \mathrm{g/s}$

(B) $2.25\ \mathrm{g/s}$

(C) $0.18\ \mathrm{g/s}$

(D) $0.44\ \mathrm{g/s}$

(E) $0.96\ \mathrm{g/s}$

To estimate the rate of evaporation in the constant rate period from a mass transfer point of view, what system parameters are needed? (Multiple answers)

(A) Reynolds number of liquid water.

(B) Temperature of the water.

(C) Altitude.

(D) Relative humidity of the air.

(E) Rayleigh number of liquid water.

Based on the above information, what would the feed condition be?

(A) Superheated vapor

(B) Superheated liquid

(C) Subcooled vapor

(D) Subcooled liquid

(E) Mixture of liquid and vapor

The distillation column is operated at 1.5 times the minimum reflux ratio. How many ideal plates are at least needed to achieve the design specifications?

(A) 3

(B) 5

(C) 7

(D) 9

(E) 11

With the same reflux ratio as in Question 22, the feed condition has changed to saturated liquid. Please estimate how many plates can be reduced or should be added to achieve the design specifications.

(A) 1 reduced

(B) 2 reduced

(C) 1 added

(D) 2 added

(E) 3 added

A heat exchanger was designed to heat a process liquid stream from $40^\circ\mathrm{C}$ to $80^\circ\mathrm{C}$ using saturated steam at $100^\circ\mathrm{C}$. The heat exchanger is operated at the designed conditions with the designed exchanger area and specifications. One day, the flow rate of the liquid is reduced by 50% due to a process upset. What do you expect the outlet temperature of the liquid stream to be if the inlet temperature of the process liquid is unchanged?

Please estimate under the following assumptions:

• The overall heat transfer coefficient is dominated by the process liquid side and can be estimated by $U=C\mathrm{Re}^{0.8}\mathrm{Pr}^{0.33}$. Here, $U$ is the overall heat transfer coefficient of the heat exchanger; $C$ is a constant; $\mathrm{Re}$ is the Reynolds number; and $\mathrm{Pr}$ is the Prandtl number.
• All the physical properties of the liquid stream are temperature-independent.
• The process liquid stream is in a single phase.
• The saturated steam side is at a constant temperature of $100^\circ\mathrm{C}$.

(A) $77^\circ\mathrm{C}$

(B) $80^\circ\mathrm{C}$

(C) $83^\circ\mathrm{C}$

(D) $86^\circ\mathrm{C}$

(E) $89^\circ\mathrm{C}$

A fluid flows (along the $z$ direction) through a slightly tapered tube with length $L$. The tube radius changes linearly from $R_0$ at the tube entrance to a slightly smaller value $R_L$ at the tube exit. Assume that the Hagen–Poiseuille equation is approximately valid over a differential length $dz$ of the tube so that the mass flow rate is

where $\rho$ and $\mu$ are the density and viscosity of the fluid, respectively, and $P$ is the modified pressure.

(a) Write down the assumptions that were made in obtaining the Hagen–Poiseuille equation. (5%)

(b) Write down the expression for $R$ as a function of $z$ for the tapered tube. (4%)

(c) Change the independent variable in Eq. (1) to $R$, so that the equation becomes

(5%)

(d) Integrate Eq. (2) to obtain an expression for the mass flow rate. (6%)

Please estimate the total number concentration of silica nanoparticles in the aqueous phase (unit: particles/$\mathrm{cm^3}$ of solution).

What are the retention time of the particle in the spray dryer and the average drying rate in this system, in terms of $\mathrm{kg}$ of water per minute?

Based on (a) and (b), the particle concentration in the aqueous phase can be effectively increased by up to 20 times using a centrifugal cleaning method, i.e., a mechanical separation process. What is the maximum total number concentration of silica nanoparticles in the aqueous phase after the centrifugation process?

Instead of spray drying, a regular plate dryer is used to dry the concentrated solution with a particle mass concentration of 20.0%. Due to the significant increase in particle concentration, the critical moisture content must be considered. Assume that the critical moisture content is $4\ \mathrm{kg\ H_2O/kg\ silica\ nanoparticle}$ and that the drying rate decreases linearly with water content. The drying area is $1.0\ \mathrm{m^2}$, and the drying rate at the critical point is $0.02\ \mathrm{kg\ H_2O/(m^2\cdot min)}$.

Would you expect complete drying within 5 hours of operation, where complete drying is defined as a free moisture content of less than $0.00001\ \mathrm{kg\ H_2O/kg\ silica\ nanoparticle}$?

Draw a breakthrough curve in terms of the relative/normalized number concentration of airborne virus particles eluting from the cylindrical filter versus the time of operation.

Estimate the saturation capacity of the adsorbent in terms of the number of virus particles per gram of adsorbent.