Homework Set 4. gas B open end

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1 Homework Set 4 (1). A steady-state Arnold ell is used to determine the diffusivity of toluene (speies A) in air (speies B) at 298 K and 1 atm. If the diffusivity is DAB = m 2 /s = 8.44 x 10-6 m 2 /s, and the ell has a 0.8 m 2 ross-setional area and a onstant diffusion path length of 10 m, how muh toluene in units of m 3 per hour must be supplied to the ell to maintain a onstant liquid level? At 298K, the vapor pressure of toluene is 28.4 mmhg and its speifi gravity is gas B open end N Az (z=z+ z) N Az (z=z) z = z 2 z z = z 1 pure liquid A liquid-gas interfae Note that onentration of toluene in the air blowing past the top of the Arnold ell is effetively zero as it is extremely dilute. The molar mass of toluene is g/mol. (2). A spherial ball of solid, nonporous naphthalene, a moth ball, is suspended in still air. The naphthalene ball slowly sublimes, releasing the naphthalene into the surrounding air by a diffusion limited proess. Estimate the time required to redue the diameter from 2 m to 0.5 m when the surrounding air is at 347 K and Pa. Naphthalene has a moleular weight of 128 g/mole, a solid density of g/m 3, a diffusion oeffiient in air of m 2 /s, and exerts a vapor pressure of 5 torr (670 Pa) at 347 K. While this problem is not stritly steady state, for purposes of the approximate alulation you may treat it as steady state. (3). In the manufature of semionduting thin films, a thin film of solid arseni is laid down onto the surfae of a silion wafer by the hemial vapor deposition of arseni, AsH3

2 2AsH3(g) 2 As(s) + 3 H2(g) The arseni atoms then diffuse into the solid silion to dope the wafer. What is the flux of arseni atoms into the silion wafer after one hour, in units of atoms/m 2 s? What is the arseni onentration 2 mirons into the silion wafer after one hour, in units of atoms/m 3? The initial onentration of residual arseni in the silion wafer is atoms/m 3. The proess temperature is 1050 C. The average diffusivity of arseni in silion is m 2 /s at this temperature, and its maximum solubility in silion is atoms/m 3. The Si wafer an be onsidered to be semi-infinite. Also, tabulated values of the error funtion an be found at (4). A 750 Watt immersion heater in the form of a ylinder with a 0.75 inh diameter and 6 inhes in length is plaed in 20 o C stagnant water. The ylinder is oriented with its axis horizontal. Heating of the water produes free (natural) onvetion of water around the ylinder. This free onvetion removes heat from the ylinder's surfae at a rate that an be alulated with the help of the following orrelation established from experimental data: Nu = 1/ Ra /16 1 (0.559/ Pr) 2 8 / 27 This orrelation was derived for ylinders with a high aspet ratio (as in the present problem), immersed in a fluid reservoir and oriented horizontal (that is, perpendiular to the gravitational field). Nu is the Nusselt number, Pr is the Prandtl number, and Ra is the Rayleigh number. For this situation, these dimensionless groups are defined as follows: Nu = hd/ Pr = ĉ p / Ra = D 3 g T / ( ) D is the ylinder radius, g is the gravitational onstant, 0 is the density of the fluid at the bulk temperature, is the oeffiient of thermal expansion of the fluid, T is the temperature differene between surfae of the ylinder and bulk fluid, is the fluid visosity, is the thermal diffusivity of the fluid, is the thermal ondutivity of the fluid, and h is the heat transfer oeffiient desribing heat transfer from the ylinder to the surrounding fluid due to free onvetion. Ra an be shown to equal the produt of the Grashof and Prandtl numbers, Ra = Gr Pr. The produt T equals the differene in density between fluid at the ylinder surfae and in the bulk. In the above expressions, the properties of the fluid (other than 0) are evaluated at the surfae temperature of the ylinder.

3 Calulate the surfae temperature of the ylinder, given that steady state applies (Note: a trial and error solution is required). Properties of water as a funtion of temperature an be found at (5). A ooking oven has a top surfae temperature of 45 C when exposed to still air. At this ondition the inside oven temperature and room air temperature are 180 C and 20 C, respetively, and heat is transferred from the top surfae at 40 W. To redue the surfae temperature, as required by safety regulations, room air is blown aross the top with a veloity of 20 m/s, whih is measured to redue the surfae temperature to 24 C. What will be the rate of heat transfer from the top surfae under this new operating ondition? When the room air is still (initial situation), the heat transfer from the oven top to the air is by natural onvetion only. Under this situation we an take NuL = 0.15 Ra 1/3 (works for 10 7 < Ra < ) Assume the shape of the oven top is square, with a length L on eah side (you'll need to figure out what L is from the given information note that the total rate of heat transfer is given in Watts, rather than a heat flux). For the fored onvetion ase when air is blown aross the top of the oven, there are several options of orrelations. Two are reprodued below (the laminar flow orrelation follows from boundary layer theory) NuL = Re L 1/2 Pr 1/3 (works for laminar flow, ReL < ) NuL = (0.037 Re L 4/5 871)Pr 1/3 (works for transition flow, < ReL < ) You an assume that the heat transfer from the top of the oven is dominated by fored onvetion (i.e. the fored onvetion heat transfer is muh larger than any residual natural onvetion transfer). In these orrelations the properties of air are to be evaluated at the film temperature Tf, whih is the average of the surfae and bulk temperatures. Properties of air are at Note that the internal oven temperature of 180 C is not needed in this problem. The subsript L indiates to use L as the harateristi dimension of the problem. (6). Trihloroethane, Cl3CCH3 (TCA), is used to hlorinate films of SiO2 grown by thermal oxidation. A semiondutor fabriation proess has been proposed involving the evaporating of the TCA into a flowing inert gas stream. A pan will ontain the liquid TCA at a starting depth of 0.01 m and a length of 4 m in the diretion that inert gas will flow at 6 m/s. The pan is quite wide. The liquid TCA will be maintained at a temperature of 293K and the system s pressure will be Pa. Under these onditions, the vapor pressure of TCA is Pa, the kinemati visosity of the inert gas is = / = m 2 /s, and the diffusivity of TCA may

4 be assumed to be m 2 /s. If the density of the liquid TCA is assumed to be 1 g/m 3 and the transition from laminar to turbulent ours at Ret = , determine the time that will be required to evaporate the TCA. The molar mass of TCA is 133 g/mol. Above it states that the transition from laminar to turbulent flow of the gas over the pan ours at Ret = For flows over a flat surfae like the pan, the flow remains laminar for short distanes but as the length of the surfae (here pan) inreases the perturbation to the flow builds and a transition from laminar to turbulent flow takes plae. This is quantified by the Reynolds number, whih is defined as Rex = Vx/ where x is the distane traveled along the surfae and V is the veloity of the flow far from the surfae, where it remains unperturbed. Thus, Rex inreases with distane from the leading edge of the pan. This means that mass transfer over the initial setion of the pan, for whih Rex < Ret, will follow the orrelation for laminar flow, while that from the remaining area of the pan where Rex > Ret will obey the orrelation for turbulent flow. The appliable orrelation for suh a ombined situation of laminar and turbulent flows an be obtained by averaging over the full length of the pan, leading to k 0.664D AB S 1/ 3 Re 1/ 2 t D L AB S 1/ 3 (Re 4 / 5 L Re 4 / 5 t ) The above equation provides an expression for an averaged mass transfer oeffiient for a plate of total length L over whih part of the flow is laminar (orresponding to first term in the numerator) and part turbulent (the seond term in the numerator). (7). Determine the value of Henry s law onstant, in Pa/mole fration of ammonia (as xnh3 0) for ammonia (NH3) in water. The following equilibrium data at 298 K were reported in the Chemial Engineering Handbook: partial pressure NH3, mmhg wt NH3/100 wt of water Take the molar mass of ammonia as g/mol. (8). An absorption tower, operating at 293 K and Pa, was used to absorb SO2 from an air mixture into water. At one point in the equipment, the partial pressure of the SO2 on the gas stream was Pa and the onentration of the ontating liquid stream was 0.55 kgmole/m 3. The individual film mass-transfer oeffiients at 293 K and Pa were

5 and kg = kgmole/m 2 s Pa kl = kgmole/m 2 s (kgmole/m 3 ) Equilibrium data at 293 K are as follows: Partial pressure SO2, Pa Conentration in kgmole/m 3 soln A kgmol is a number of partiles equal to 1000 times Avogadro's number (i.e. 1 kgmol = partiles). The plae to start for part (a) is to equate the molar fluxes on the gas and liquid sides of the interfae (assuming no aumulation at the interfae) and to make use of the equilibrium data provided. You may wish to fit the equilibrium data to a polynomial for ease of interpolation, but this is not neessary (i.e. manual interpolation between the given values is fine). (a) Evaluate the interfaial onentration, A,i, and pa,i. (b) Fill in the values for the following table, giving the various oeffiients and the assoiated driving fore. Coeffiient kg = kl = KG = KL = Driving Fore pag - pai = Ai - AL = pag - pa* = A* - AL = () What perentage of the overall mass-transfer resistane is in the gas film? (9). The outside walls of a house are onstruted of a 4-in. layer of brik, ½ in. of elotex, an air 5 spae 3 in. thik, and ¼ in. of wood paneling. If the outside surfae of the brik is at 30 F 8 and the inner surfae of the paneling at 75 F, what is the heat flux if the air spae is assumed to transfer heat by ondution only? Below are the needed thermal ondutivities: kbrik = 0.38 Btu/hr ft o F kelotex = Btu/hr ft o F kair = Btu/hr ft o F kwood = 0.12 Btu/hr ft o F