Ind. Eng. Chem. Res. 2006, 45, 3287-3292
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Measuring Moisture Content of Potash Bulk Fertilizers Using a Steel Ball in a Transient Heat Transfer Process Richard W. Evitts* and Todd Yungwirth Department of Chemical Engineering, UniVersity of Saskatchewan, 57 Campus DriVe, Saskatoon, Saskatchewan, Canada S7N 5A9
Robert W. Besant Department of Mechanical Engineering, UniVersity of Saskatchewan, 57 Campus DriVe, Saskatoon, Saskatchewan, Canada S7N 5A9
This research measures the effective thermal conductivity of a bed of granular potash particles and determines its relationship to moisture content by measuring the time dependent temperature of a heated steel spherical ball inserted into the bed. In these tests, the steel ball is heated to 5-10 °C above the bed temperature and then placed into the bed. As the heat diffuses throughout the bed, a data acquisition system records the temperature of the ball. The bed effective thermal conductivity is then calculated from these data. Tests were performed for five different ranges of particle sizes at five moisture contents (i.e., 0, 0.25, 0.5, 1.0, 2.0, and 3.0 wt %). The coefficient of determination for a linear fit between moisture content and effective thermal conductivity was found to vary from 0.913 to 0.998. This technique is expected to be practical for field measurements with granular materials, such as bulk fertilizers, because data can taken at selected points in the bed and because of its low cost, quick response, and simplicity of instrumentation. Introduction Potash is used as a plant fertilizer around the world and is often stored, transported, and distributed as a granular bulk material. It primarily consists of potassium chloride, which is also known as sylvite. Potassium is a nutrient that is required in large amounts by most plants. Traces of sodium chloride or halite (NaCl) and carnallite (KMgCl3‚6H2O) complete the chemical composition of potash. When exposed to air with a relative humidity above 50%, granular potash absorbs moisture from the air. If the duration of exposure is long enough, the particles may partially or fully dissolve causing the bulk properties of the bed to be altered. Once moisture is absorbed, any subsequent drying causes the formation of crystal bridges between the particles in the bed; this physical change is known as caking. When many crystal bridges form, the potash particles clump together. Agricultural machines are unable to distribute caked potash evenly on the land. Caking can occur in potash at moisture contents as low as 0.25 wt %, and if above 0.5 wt %, considerable loss in quality can result from caking. A low-cost method to determine the moisture content in a potash particle bed before degradation occurs could minimize the quality loss of stored bulk fertilizer. The experiments involved in this research work focus on an indirect method of measuring moisture content by first determining the effective thermal conductivity. A small spherical steel ball with an embedded thermocouple is heated to an elevated temperature (i.e., 5-10 °C above the bed temperature), which is then placed in a granular bed. The effective thermal conductivity of the potash determines the rate of cooling of the steel ball. Other properties, such as bed porosity and specific heat, also influence the rate of cooling, but these are more easily measured or known from data found in the literature. * To whom correspondence should be addressed. Tel.: (306) 9664766. Fax: (306) 966-4777. E-mail:
[email protected].
Other methods have been used to determine the moisture content of beds of granular material such as the gravimetric method and the Karl Fischer titration method. The gravimetric method involves weighing a sample, drying it completely, and then reweighing. This method is often very time-consuming. The Karl Fischer titration method is complex, costly, and has problems if the substance under investigation has bound water, such as water of crystallization. Experiments were performed with a narrow range of particle sizes to ensure that the methodology of testing and the accuracy of the tests are acceptable rather than presenting a complete data set of all potash product grade sizes. The main focus of this study is on the granular grade product with a dominant size range of 2.00-3.96 mm in diameter and on the standard grade product with a dominant size range of 0.85-1.18 mm in diameter. A final set of experiments was performed on particle size mixtures that have the same particle size distribution as the actual product grade. The detailed objectives of this study are to use this thermal conductivity probe: (1) to present data and analysis to accurately determine effective thermal conductivity and moisture content of potash test samples, each with a narrow range of particle sizes and particle packings and a known moisture content; (2) to develop a relationship between the effective thermal conductivity and the moisture content described in objective 1 and to use this relationship to determine the moisture content and test these correlations on a sample comprised of various particle sizes that form a typical potash product. Theory Porosity, , is defined as the fraction of the bulk volume of the porous sample that is occupied by pore or void space.1 The most accurate way to calculate the porosity is by the density method, and this was used in the present study. The following equation describes the relation between porosity and the densities of a potash bed:
10.1021/ie051080o CCC: $33.50 © 2006 American Chemical Society Published on Web 04/04/2006
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Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006
FB ) Fa + (1 - )Fs
(1)
where FB, the density of the bulk mixture, and Fs, the density of the solid, are measured by weighing known volumes of material. Since the density of the air, Fa, is much smaller than FB and Fs, this equation is simplified to
)1-
FB Fs
(2)
Fourier’s law of heat conduction may be applied to a porous media when the temperature differences between the particles and the interstitial air in any small volume element is negligible. In this case, the law is used, for an isotropic media, to define the effective thermal conductivity of the porous bed at any point in the bed:
ke ) -
q′′r
(3)
(∂T/∂r)
Using the thermal energy balance principle on any volume averaged element,2 the equation for heat conduction in dry porous media, locally at thermal equilibrium and without any significant free-convection effects, can be written as3
(FCp)e
∂T 1 ∂ ∂T k r2 ) ∂t r2 ∂r e ∂r
(
)
diameter (i.e., D . 0.5dp). The thermal energy storage profile inside the bed during a transient test will vary with an exponentially shaped decay curve in the radial direction at any time, and the amount of energy stored inside the first half particle diameter distance will be much smaller than the rest of the bed when the thermal disturbance reaches a thermal boundary layer thickness, δi . 0.5dp. For this condition, the boundary thermal capacitance effect will not significantly alter the steel ball temperature, and it is now possible to solve eq 4 with its boundary condition, eq 7, for the case of constant T∞, assuming that ke is constant except for the boundary condition where it is assumed that keb ) 0.55ke for the first half particle adjacent to the ball and at this same boundary condition the effective thermal diffusivity at the boundary, Reb, is assumed to be equal to the bulk effective thermal diffusivity, Re. A sensitivity study of these two assumptions showed that a 10% variation in keb and Reb caused the effective thermal conductivity, ke, to change by less than 5%. That is, provided that δi . 0.5dp, the exact values of these boundary conditions have a small impact on the bias uncertainty of ke but when this bias is constant it will not significantly effect the final moisture content results. A good approximation for the value of δi(t) is given by the expression,4
δt ) x12Ret
(4)
where the effective thermal diffusivity is defined as
Re ) ke/(FCp)e
where T is the local bed temperature, the effective thermal conductivity, ke, is the property defined in eq 3, and
(FCp)e ) (FCp)a + (1 - )(FCp)s
(5)
is the effective heat capacity term for a bed of porosity, . The Biot number, Bi, is a ratio of the resistance to heat flux in a solid of finite size to the resistance in the surrounding media of infinite extent. Provided that the Biot number is small (i.e., Bi < 0.1), the steel ball diameter, D, and its thermal properties (i.e., thermal conductivity, kst, and specific heat, Cpst) will not have any significant internal temperature differences at any time. An energy balance close to the surface of the steel ball in the bed was performed, and this produced the definition of Bi as
1 keb ∂T /(T - T∞) Bi ) D 6 kst ∂r b b
( )
(6)
where Tb is the boundary surface (steel ball) temperature and T∞ is the initial bed temperature away from the thermal disturbance of the heated ball and the boundary condition for T. The effective thermal conductivity in the particle bed adjacent to the steel ball is keb. For all the tests, Bi was much less than 0.1; therefore, the temperature of the steel ball was uniform at any time, and then, the boundary condition for the particle bed at the surface of the ball at any time becomes
dTb ∂T 1 D(FCp)st ) keb 6 dt ∂r b
( )
(8)
(7)
Studies indicate that the porosity adjacent to solid surfaces in a particle bed is much higher than that some distance away.2 However, this effect is negligible for distances from the ball greater than about one-half of a particle diameter, implying the particle porosity boundary effect will not significantly alter the effective thermal conductivity of the bed beyond one-half of a particle diameter away from the ball and also that the ball diameter should be much larger than the average particle
(9)
Equation 4, for the temperature distribution inside the particle bed, and its boundary condition, eq 7, are set up in finite difference form and can be solved with any given effective thermal conductivity, ke. The best value of ke for any experiment will be the one that satisfies these equations and gives the best fit to the experimental data for the temperature-time behavior of the steel ball. The method used to determine the best value of ke is discussed below. Data Analysis Transient temperature data was recorded for many samples of potash with the same particle size range and moisture content. For each test, the best value of the effective thermal conductivity of the bed was determined using the parameter estimation method described below. The effective thermal conductivity of the bed was determined using the Gauss-Newton method for algebraic models. This method is used to estimate an unknown parameter, in this case ke, that appears in a model described by a set of algebraic equations, given as eq 10.
yj ) Cf (xj,khe)
(10)
The structure of the mathematical model and the objective function to be minimized is known. The model is described as follows. The effective thermal conductivity is represented as a vector of parameters that are unknown. The independent variables are represented by the vector xj, which represents time, and yj is the temperature of the steel ball. The vector function, Cf, is a vector in the form of algebraic equations. The data obtained in the experiment is in the form [yi,xi], i ) 1, ..., N, where N data points have been collected in an experiment. The model’s predictions must match these data. The objective function that needs to be minimized is based on the statistical properties of the experimental error involved in the
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measurement of yj, the output vector. It expresses the overall departure of the values calculated by the model to those of the experimental measurements. This objective function, which is to be minimized, can be written as
S(ke(j) + µ∆k(j+1)) < S(ke(j))
(18)
Finally, a convergence criterion is defined which will minimize iterations for a selected convergence criteria.
N
S(ke) )
eTi ei ∑ i)1
(11)
The residual, ei, in eq 11 is the individual measurement departure from the model calculated value. For the ith residual, the explicit algebraic model is
ei ) [yi - f(xi,ke)]
(12)
where the model based value of f(xi,ke) is calculated using the estimated parameter values. The Gauss-Newton method is used to minimize the value of S(ke) because it has been proven to be one of the most efficient methods for estimating parameters in nonlinear models.5,6 This method begins with the estimation ke(j). Linearization of the model equations around ke(j) yields the following equation:
f(xi,ke(j + 1)) ) f(xi,ke(j)) +
( )
∂fT T (j + 1) ∆ke + HOT; ∂ke i ) 1, ..., N (13)
By neglecting all of the higher order terms (HOT), the model output at k(j+1) can be approximated by
y(xi,kej+1) ) y(xi,ke(j)) + Gi∆ke(j+1); i ) 1, ..., N (14) where Gi is a sensitivity matrix, (∂fT/∂f)Ti ) (∇fT)Ti , evaluated at xi and k(j). Substitution of y(xi,kj+1), as approximated by the above equation, into the least squares objective function and use of the critical point criterion
∂S(ke(j+1)) ∂ke(j+1)
)0
(15)
yields a linear equation in the following form:
A∆ke(j+1) ) b
(16)
where N
A)
GTi Gi ∑ i)1
N
b)
GTi [yi - f(xi,ke(j))] ∑ i)1
Equation 16 can be solved to yield, ∆ke(j+1). This is used to estimate the next value of k via
ke(j+1) ) ke(j) + µ∆ke(j+1)
(17)
The relaxation stepping parameter, µ (0 < µ
