2 Units and Dimensions
17. Sherwood number: It is represents a dimensionless mass transfer coefficient and is the ratio of concentration gradient in the fluid in contact with a solid wall to a reference concentration gradient
applicable to the entire fluid.
kLSh =C DAB
where kC is the mass transfer coefficient, L is the characteristic linear dimension for the flow system, and DAB is the mass diffusivity.
18. Stanton number: The Stanton number for heat transfer (StH) is the ratio of the Nusselt number to the product of the Reynolds number and the Prandtl number while the Stanton number for mass
transfer (StD) is the ratio of the Sherwood number to the product of the Reynolds number and the Schmidt number.
hStH = CG
P
kCStD =v
where G is the mass velocity (rv), h is the heat transfer coefficient, CP is the heat capacity, kC is the mass transfer coefficient and v is the velocity.
EXERCISES
2.1 The thermal conductivity of an insulating brick is 0.15 Btu/(ft h °F). Express the thermal conductivity in SI units.
2.2 The heat transfer coefficient of oil flowing through a pipe is 300 W/(m2 K). Determine the value of heat transfer coefficient expressed in
(a) kcal/(h m2 °C) (b) Btu/(h ft2 °F)
2.3 Thermal conductivity of pure iron is 39 Btu/(ft h °F ) and that of steel containing 1% C is 39 kcal/(m h °C). Which one is a good conductor of heat, iron or steel?
2.4 The viscosity of water is one centipoise. Express the viscosity of water in the following units?
(a) Pa s (b) lb/(ft h)
2.5 The mass velocity of a gas through a duct is 1000 kg/m2 h. Express the velocity into lb/ft2 s.
2.6 A man circling the earth in a spaceship weighed 300 N at a location where the local gravitational acceleration was 4.5 m/s2. Calculate the mass of the man and his weight on the earth where the gravitational acceleration is 9.81 m/s2.
2.7 Convert 1 bar into equivalent kgf/cm2 and psi.
2.8 A special manometer fluid has a specific gravity of 2.95 and is used to measure a pressure of 1.15 bar at a location where the barometric pressure is 760 mm Hg. What height will the manometer fluid indicate?
2.9 The potential energy of a body of mass 10.0 kg is 1.5 kJ. What is the height of the body from the ground? If a body of mass 10 kg is moving at a velocity of 50 m/s, what is its kinetic energy?
2.10 A man whose weight is 600 N takes 2 min for climbing up a staircase. What is the power developed in him, if the staircase is made up of 20 stairs, each 0.18 m in height?
2.11 Nitrogen gas is confined in a cylinder and the pressure of the gas is maintained by a weight placed on the piston. The mass of the piston and the weight together is 50 kg. The acceleration due to gravity is 9.81 m/s2 and the atmospheric pressure is 1.013 25 bar. Assume frictionless piston.
Determine:
(a) The force exerted by the atmosphere, the piston and the weight on the gas if the piston is 100 mm in diameter.
(b) The pressure of the gas.
(c) If the gas is allowed to expand pushing up the piston and the weight by 400 mm, what is the work done by the gas in kJ?
(d) What is the change in the potential energy of the piston and the weight after the expansion in part (c)?
2.12 A spherical balloon of diameter 0.5 m contains a gas at 1 bar and 300 K. The gas is heated and the balloon is allowed to expand. The pressure inside the balloon is found to vary linearly with the
diameter. What would be the work done by the gas when the pressure inside reaches 5 bar?
2.13 The diffusivity of a gas pair A-B is given by
⎛⎞
1/ 2
1.8583 10
–3 3 / 2
×+
D
AB
=
T MM⎜⎟⎝⎠
P s2W
where DAB is in cm2/s, T is the absolute temperature in K, MA and MB are molecular weights of species A and B, respectively, P is the pressure in atm, sis the Lennard– Jones parameter in angstrom units and W is a dimensionless parameter. If the equation is written as
3/ 2 ⎛⎞1/ 2 D
AB
=
CT MM⎜⎟⎝⎠
P s2W
where DAB is in m2/s, T is in K, P is the pressure in Pa, sis the Lennard–Jones parameter in nanometres, what is the value of C?
2.14 For fluids in turbulent motion through tubes, the heat transfer coefficient is given as ⎛⎞
0.8
ha=⎜⎟⎜⎟⎝⎠
The numerical value of the constant a = 10.1, when h is measured in Btu/(ft2 h °F), CP is the specific heat of the fluid given in Btu/(lb °F), G is the mass velocity in lb/(ft2 s) and the diameter is in ft.
Determine the value of a (a) when h is measured in W/(m2 K), CP is the specific heat of the fluid given in kJ/(kg K), G is the mass velocity in kg/(m2 s) and (b) when h is measured in kcal/(m2 h °C), CP is the specific heat of the fluid given in kcal/(kg °C), G is the mass velocity in kg/(m2 s) and the diameter in m.
2.15 The heat capacity of carbon dioxide gas is given by CP = 0.1978 + 1.059 ¥ 10–4t – 2.395 ¥ 10–8t2
where CP is in Btu/(lb°F) and t is in °F. Change the equation into the form in which CP is given in kJ/(kmol K) and temperature is in K.
2.16 The following equation relates the heat capacity of CO gas with temperature. CP = 26.586 + 7.582
¥ 10–3T – 1.12 ¥ 10–6T2
where CP is expressed in kJ/kmol K and T in K. Obtain equations which will express (a) The heat capacity in cal/mol K, temperature in K.
(b) The heat capacity in cal/mol °C, temperature in °C.
(c) The heat capacity in J/kg K, temperature in K.
(d) The heat capacity in Btu/lb-mol °F, temperature in °F.
2.17 The Antoine equation predicts the effect of temperature on vapour pressure by the relation:
ln P
s = A –B TC
where A, B, and C are Antoine constants. For water, the values of the constants when the vapour
pressure is in kPa and temperature is in K are found to be A = 16.26205, B = 3799.887 and C = 46.854.
What are these values when
(a) Vapour pressure is in mm Hg and temperature in °C.
(b) Vapour pressure is in psia and temperature in °F.
2.18 The heat capacity of sulphuric acid in Btu/(lb-mol °R) is given by CP = 23.06 + 2.071 ¥ 10–2T where T is in °R. What is the equivalent expression if heat capacity is given in (a) kJ/(kmol K) and T is in K (b) kcal/(kmol °C) and temperature in °C. 2.19 Stefan–Boltzmann law states that the emissive power is proportional to the fourth power of absolute temperature. Stated mathematically,
Eb = sT 4
where Eb is the emissive power in Btu/(ft2 h), sis the Stefan-Boltzmann constant, and T is the temperature in °R. What is the value of sin W/(m2 K4), if its value in FPS units is 0.171 ¥ 10–8 Btu/(ft2 h °R4)?
2.20 Mass flow rate is the product of volumetric flow rate and density. If the volumetric flow rate of a liquid of specific gravity 0.78 is 100 ft3/min, find the flow rate in kg/s. 2.21 The following empirical equation was proposed to estimate the rate of heat loss from the walls of a cylindrical pipe to the ambient air:
D T
1.25
q =0.5D0.25
where q is the rate of heat loss in Btu/ft2 h, DT is the temperature difference in °F and D is the diameter in inches.
(a) Determine the units of the constant.
(b) Express the equation in SI units.
2.22 Rate of a gas phase reaction (A¤ B) taking place at 25°C and 1 atm is given by
dC 2.2 k p0.5––A ABdt =
with k1= 17.3 and k2 = 0.2, where CA is the concentration in kmol/m3, t is the time in minutes, and pA, pB are partial pressures in atm.
(a) Find the units of k1 and k2.
(b) If the rate is to be written in the form of
dpA AB0.5––
dt
⎛⎞pi
what are the values of k¢1 and k¢2 ⎜⎟⎝⎠
2.23 The van der Waals equation of state is given as
⎛⎞aPVbRT⎜⎟⎝⎠+−=
where a and b are known as the van der Waals constants. For CO2 the constants are found to be a = 0.365 N m4/mol2 and b = 4.28 ¥ 10–5 m3/mol. If the pressure is given in atm and the molar volume in L/mol, what are the values of the constants?
2.24 The volume of a microbial culture is observed to increase according to the relation V (cm3) = et where t is in seconds. Find the equivalent expression where V is in inch3 and t is in hours.
2.25 The pressure drop for a fluid flowing through a packed bed of solids is given by the relation:
0.15 0.85 1.85
D
⎡⎤mr Hv
P=⎢⎥⎢⎥⎣⎦
where DP is pressure drop in Pa, mis fluid viscosity in poise, H is the bed depth in metres, ris fluid density in kg/m3, v is fluid velocity in m/s and DP is packing diameter in metres. Modify the relation such that the pressure drop is still in Pa, but other terms are in FPS units.